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Constitutive models for regression of various experimental stress–strain relations
International Journal of Mechanical Sciences 101-102 (2015) 1–9
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Constitutive models for regression of various experimental stress–strain relations Weilong Hu a, Yanli Lin b,n, Shijian Yuan c, Zhubin He c a
Troy Design & Manufacturing Co., 25111 Glendale, Redford, MI 48239, USA School of Materials Science & Engineering, Harbin Institute of Technology at Weihai, Weihai 264209, PR China c School of Materials Science & Engineering, Harbin Institute of Technology, Harbin 150001, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 2 March 2015 Received in revised form 16 June 2015 Accepted 10 July 2015 Available online 23 July 2015
Any of a theoretical constitutive model all needs to be characterized by proper experimental stress– strain relations to describe material behaviors in the domain of plastic deformation. It means that regression of experimental data by a suitable equation would be the most fundamental work. Some empirical equations currently used in engineering cannot always reproduce experimental stress–strain relations acceptably, particularly for experiments dealing with complex stress states and multiexperimental stress–strain relations. To improve the accuracy of numerical analysis, two new constitutive models for regression of various experimental stress–strain relations are suggested. One is simple relatively to another, but the complex one can give a more accurate reproduction of experimental stress– strain relation. Several uniaxial tensions of rolled sheet metals and a hydro-bulging test have been performed. Regressions of these experimental stress–strain relations have been done by two suggested models. Predicting precisions are discussed and compared with power functions. In relation to the uniaxial tension in different loading direction to the rolling, changeable ratios of transverse to through thickness strain-increments are explained, and their influence on the profile of plastic potential is presented. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Experimental stress–strain relation Constitutive model Regression Anisotropy Rolled sheet metal
1. Introduction Constructing an appropriate constitutive model is a critical step for obtaining a reasonable simulation result regarding a forming process. Experimental stress–strain relations dealing with different stress states are the most natural representation of material properties. Any of a constructed constitutive model all needs to be characterized by some experimental stress–strain relations. Capability of a constitutive model completely depends on the experimental data addressed in the model. Because of the rapid development of computer technology, numerical analyses have become the major method in engineering applications, and simulating accuracy has become the most important consideration in development of a new theory. Currently, many simplified models used to reproduce an experimental stress–strain relation, such as perfect rigid-plastic model and linear strain-hardening model etc., are seldom adopted in numerical analyses. A more accurate equation for reproducing experimental stress–strain relation is requested. If a real case could be well predicted by a constitutive model, the key issue would be that whether this model could
n
Corresponding author. Tel./fax: þ 86 631 5677408. E-mail address: [email protected] (Y. Lin).
http://dx.doi.org/10.1016/j.ijmecsci.2015.07.010 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
describe mechanical properties of materials good enough. It implies that multiple effects observed in different experiments could be accurately taken into account in the constitutive model. In relation to rolled sheet metals, experimental data include experimental stress–strain relations of uniaxial tension dealing with different loading direction, and they are used mainly to describe the anisotropic hardening of rolled sheet metals [1–6]. Some experimental data relate to loading path change that may vary the feature of original experimental stress–strain relations, such as the phenomena of plastic spin [7,8]. With respect to experimental stress– strain relations dealing with loading and reverted loading process, they are used to describe the Bauschinger effect [9–11]. Predicting accuracy of all constitutive models would basically rely on these original experimental stress–strain relations. It denotes that proper regressions of experimental stress–strain relations would be a more basic requirement to ensure that constitutive models could respond to an acceptable precision in predictions. If these original experimental stress–strain relations cannot be treated close to their realities, a proposed theory would significantly lose its function in description of a real deformation process. Because various materials present rather different mechanical properties, their stress–strain relations arising from different stress states cannot be defined all following a similar empirical formula, such as the most often used power law. Just dealing with a simple uniaxial
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tension, if two experimental stress–strain relations of a rolled sheet metal all need to be used in describing a changeable ratio of transverse to through thickness strain-increments, these two experimental stress–strain relations cannot be always correctly reproduced by two power expressions. However, these two experimental stress–strain relations are very important factors used to present the anisotropic flow behavior of rolled sheet metals. To solve this problem, we may directly use the experimental data in a theoretical constitutive model. However, if several experimental stress–strain relations all need to be addressed in this constitutive model to describe anisotropic hardening behavior of rolled sheet metals, equivalent increments of stress and strain components dealing with different experiments are also hard to be defined. If we insist on using an empirical formula, we may take a risk of some degree of the deviation between the experimental stress– strain relation and the reproduced stress–strain relation. For some cases, such deviation is actually unacceptable, particularly for the case that analyzing results are very sensitive to the hardening behavior of experimental stress–strain relation, for example the spring-back simulation of a stamped panel. No matter how many effects are introduced in a constitutive model, original experimental stress– strain relations would always be the first scale to affect its predicting accuracy. In order to obtain a more accurate reproduction of experimental stress–strain relation, two new formulations are introduced in this paper. They can be used as constitutive models for regression of various experimental stress–strain relations. Based on these models, a formulaic experimental stress–strain relation can be easily obtained to fit an acceptable accuracy. We can base on three defined points of experimental data, for example the initial yield, the maximum stress point and a point between the initial yield and the maximum stress point, to generate a constitutive model of experimental stress–strain relation. If we would like to obtain a more accurate reproduction of experimental stress–strain relation, we can also use five defined points regarding the experimental data to generate a formula of experimental stress–strain relation. By using these models, many benefits could be obtained. We can directly determine a formula of experimental stress–strain relation and do not need to clean up the serrate shape of the original testing data. It would save lots work. No matter what a stress state of the stress–strain relation is related, they all can be treated by the same constitutive model and meet an accurate requirement. Several experimental stress–strain relations are discussed to show the use of these proposed constitutive models.
2. Constitutive models of experimental stress–strain relation In order to make regression of experimental stress–strain relation more close to the reality, we attempt to propose a new constitutive model. With respect to a uniaxial tension, it deals with two experimental stress–strain relations. When doing regressions of these two experimental stress–strain relations, this model does not need all original experimental data. That is, only several points on the experimental stress–strain relation are selected to do its regression. These data can be the initial yield point, the maximum stress point and some points between the initial yield and the maximum stress, for example 5%, 10%, 15%, or 20% strains and so on. 2.1. Constitutive model with a second order function To define such a constitutive model to match an experimental stress–strain relation, at least, we need three points to generate a curve as a reproduction of this experimental stress–strain relation. We may select the initial yield point, the maximum stress point and a point A between these two points, such as the 5% strain point, or some other point with different strain value. It means
that a suitable function would have three pending coefficients to address these three experimental points. We may propose a function as a form 2 X 1 ðσ max σ Þ2 þ X 2 ε εy ðσ max σ Þ þ X 3 ε εy 1 ¼ 0 ð1Þ where σ max is the maximum stress, εy is the initial yield strain, and X 1 ; X 2 ; X 3 are the pending coefficients to address the experimental data. Therefore, these three coefficients can be obtained as 1
X1 ¼
;
X3 ¼
1
σ max σ y 2 εmax εy 2 " 2 # εA εy σ max σ A 2 1 X2 ¼ 1 εmax εy σ max σ y ðσ max σ A Þ εA εy
ð2Þ
where σ y ; εmax are the initial yield stress and the maximum strain, and σ A ; εA are the stress and strain dealing with the point A. Therefore, Eq. (1) can be finally represented as " # σÞ εA εy 2 ðσ max σ Þ2 σ max σ A 2 ε εy ðσ max þ 1 εmax εy σ max σ y ðσ max σ A Þ εA εy σ max σ y 2
ε εy 2 1 ¼ 0 εmax εy 2
ð3Þ
þ
Since this is a second order function, we can directly obtain the relationship between the stress and the strain as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i X2 X 22 1h σ ¼ σ max þ ε εy ε εy 2 X 3 ε εy 2 1 ð4Þ 2 X 2X 1 1 4X 1
ε ¼ εy
X2 ðσ max σ Þ þ 2X 3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i X 22 1h ðσ max σ Þ2 X 1 ðσ max σ Þ2 1 2 X3 4X 3 ð5Þ
We can also use another method to obtain the correlation between the stress and the strain, i.e. Eq. (1) can be given by a form as 2 ε εy 2 X 1 ðσ max σ2Þ þ X 2 ðσðεmax εy σÞ Þ þ X 3 ¼ 1 ðε εy Þ
ε εy ¼
1 X 1 E2n þX 2 En þ X 3
1=2
σ max σ ; σ max σ ¼ En ε εy ε εy σ ¼ σ max En ε εy En ¼
ð6Þ
Since we more often use a numerical method to analyze a real process, based on Eq. (1), we can get a relationship between the stress increment and the strain increment as X ðσ σ Þ þ 2X 3 ε εy Δε Δσ ¼ 2 max ð7Þ 2X 1 ðσ max σ Þ þX 2 ε εy Actually, we can also base on Eqs. (4) and (5) to obtain an incremental relationship between stress and strain. According to this constitutive model, we can get reproductions of these experimental stress–strain relations. Since this constitutive model only requires three points to determine the profile of experimental stress–strain relation, we may vary the point A to get a better fit of experimental data. For this case, strain and stress values of three selected points are 0.0019 ( 0.000867), 224.178 as the initial yield point, 0.0612 ( 0.0294), 410.7901 as the middle point, and 0.24461 ( 0.11817), 547.9195 as the maximum stress point, where values in the brackets mean the strains in transverse direction. We can use these data to determine three coefficients in the constitutive model of Eq. (1). Finally, two stress–strain relations predicted by this constitutive model are shown in Fig. 1. In relation to the stress–strain relation in transverse direction, we still use a
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
3
base on Eq. (5) to obtain relations of strain-increment as 2 3
2 4X X X ð σ σ Þ max 3 1 6 7 1 2 7 Δεt ¼ 6 h i52X Δσ 4X 2 þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 2 X 2 ðσ max σ Þ 4X 3 X 1 ðσ max σ Þ 1 2
3
2 ^ ^ ^ 4 X X ð σ σ Þ X max 6 3 1 7 1 2 Δεw ¼ 6X^ 2 þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 4 h i5 ^ Δσ 2 2X 3 2 2 X^ 2 ðσ max σ Þ 4X^ 3 X^ 1 ðσ max σ Þ 1
ð9Þ
Fig. 1. Experimental stress–strain relations reproduced constitutive model.
by the suggested
where, X^ 1 ; X^ 2 ; X^ 3 are coefficients determined by the experimental strain and stress values in transverse direction, and all strain values used are positive, but not negative values. Substituting relations of Eq. (9) into Eq. (8), we can get the ratio of transverse to through thickness strain-increments. Fig. 4 displays the distribution of this ratio varying with strain value in tensile direction. To compare it with the result obtained based on two power functions, we have
Δεt ¼
1 1=n
nk
σ tð1 nÞ=n ;
Δεw ¼
1 n^ k^
1=n^
σ ðt
1 n^ Þ=n^
1=n
R¼
Fig. 2. Experimental stress–strain relations reproduced by two power functions.
positive strain value to determine its stress value, i.e. we do not need to change the representation of this constitutive model. From this picture, predictions seem well to fit the distributions of experimental data. In order to evaluate its predicting accuracy, we may compare it with the prediction of a power function. Fig. 2 shows two stress–strain relations predicted by two power functions. Table 1 gives a summary of these two power functions dealing with the regressions of experimental stress–strain relations. Fig. 3 presents deviation analysis. Based on this comparison, predicting accuracy of this suggested function looks much better than the power function. After we get constitutive models of experimental stress–strain relations dealing with both stretching and transverse directions, we can now obtain a relation of anisotropic parameter (R-value) distributed in different hardening state. Because of the existence of the anisotropy, this parameter cannot be a constant value. Capability of representing this behavior of rolled sheet metals would give much benefit on a theoretical constitutive model for simulating a stamping process, including the spring-back prediction. Suppose that material element is an uncompressible body, we have R¼
Δεw Δεw 1 1 ¼ ¼ ¼ 1 þ Δεt =Δεw Δεt = Δεw 1 Δεz Δεw þ Δεt
ð8Þ
where Δεt ; Δεw ; Δεz are the strain-increments in stretching, transverse and through thickness directions respectively. We may
1 nk ð1 nÞ=n ð1 n^ Þ=n^ 1=n σ½ nk ¼ Δεt = Δεw 1 n^ k^ 1=n^ t
ð10Þ
^ n^ are material parameters involving the power where, k; n; k; functions. Two calculation results are also shown in Fig. 4. According to this result we can see that the ratio of transverse to through thickness strain-increments is varied with a progressive hardening process. It means that if we use a constant value of the ratio in a constitutive model, like the use currently in stamping simulations, it should affect the predicting accuracy, particularly for the spring-back prediction. One thing that needs to be paid attention is that the testing data of Rvalue expressed by strain-increments are very unstable. It means that regression precision of R-value would completely depend on two regression functions of experimental stress–strain relations. We cannot obtain an acceptable regression of R-value directly from testing data. The testing data of R-value shown in Fig. 4 were measured by each calculation step around 3% tensile strain-increments. Generally, this model can well reproduce experimental stress– strain relation of rolled sheet metals dealing with any of a testing stress states. However, if we would like to obtain a more accurate regression of experimental stress–strain relation, we may use another constitutive model suggested as follows. 2.2. Constitutive model with a fourth order function To obtain a higher accurate reproduction of experimental stress–strain relation, we may need a constitutive model that could take more experimental data into account to improve its predicting precision. With respect to this purpose, we now propose a new constitutive model for regression of experimental stress–strain relations, i.e. 2 X 1 ðσ max σ Þ4 þ X 2 ðσ max σ Þ3 ε εy þX 3 ðσ max σ Þ2 ε εy 3 4 þ X 4 ðσ max σ Þ ε εy þ X 5 ε εy ¼ 1 ð11Þ This is a fourth order function. Since this function has five pending coefficients, it can address more experimental data finely to define its characteristic in prediction of experimental stress– strain relations. We can select five points from the data of experimental stress–strain relation to characterize this function. They are the initial yield point, maximum stress point and three points A; B; C between the initial and maximum stress points.
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Table 1 Summary of two power functions. (a) Regression of a power expression in tensile direction K
n
Value 754.76731
D
Standard error 1.45603
DF Regression 2 Residual 1531 Uncorrected total 1533 Corrected total 1532 (b) Regression of a power expression in transverse direction D
K
F F
Statistics
Value 0.22019
Standard error 8.57106E 4
Reduced Chi-Square 160.65067
Adj. R-Square 0.98861
Sum of squares 3.11417E8 245956.16937 3.11663E8 2.16043E7
Mean square 1.55708E8 160.65067
F value 969235.55591
Prob4F 0
n
Statistics
Value 907.64839
Standard error 3.26731
Value 0.22978
Standard error 0.00124
Reduced Chi-Square 306.65415
Adj. R-Square 0.97726
Regression Residual Uncorrected total Corrected total
DF 2 1465 1467 1466
Sum of squares 3.01417E8 449248.32331 3.01866E8 1.97654E7
Mean square 1.50708E8 306.65415
F value 491460.63464
Prob4F 0
Fig. 4. Changeable ratio of transverse to through thickness strain-increments.
stresses of a stress–strain relation between the initial and maximum stress points. Substituting stress and strain data of these five points into Eq. (11), we finally obtain X1 ¼
1
σ max σ y
Fig. 3. Deviation analysis between the suggested function and the power function. (a) Total fitting deviation dealing with experimental stress–strain relation in tensile direction. (b) Total fitting deviation dealing with experimental stress–strain relation in transverse direction.
Suppose that these yielding stresses and strains are represented by EnðAÞ ¼
σ max σ A ; εA εy
EnðBÞ ¼
σ max σ B ; εB εy
EnðC Þ ¼
σ max σ C εC εy
ð12Þ
where, εA ; εB ; εC ; σ A ; σ B ; σ C are the experimental strains and
4 ;
X5 ¼
1
εmax εy
4
EnðAÞ 1 1 1 X4 2 þ X 5 3 X 1 EnðAÞ EnðAÞ EnðAÞ ðσ max σ A Þ4 EnðAÞ " ! EnðAÞ EnðBÞ 1 1 X3 ¼ þ X 5 ðσ max σ B Þ4 ðσ max σ A Þ4 E3nðAÞ E3nðBÞ EnðAÞ EnðBÞ 1 1 X 4 þ þX 1 EnðAÞ EnðBÞ EnðBÞ EnðAÞ EnðAÞ EnðBÞ " ! EnðAÞ EnðC Þ 1 1 þ X X4 ¼ 5 ðσ max σ C Þ4 ðσ max σ A Þ4 E3nðAÞ E3nðC Þ X2 ¼ X3
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
5
Fig. 5. Fine reproduction of experimental stress–strain relation by a more accurate model.
EnðAÞ EnðBÞ E2nðC Þ þ X 1 EnðAÞ EnðC Þ EnðAÞ EnðC Þ EnðBÞ EnðC Þ " ! EnðAÞ EnðBÞ 1 1 þ X5 3 3 ðσ max σ B Þ4 ðσ max σ A Þ4 EnðAÞ EnðBÞ EnðAÞ E2nðBÞ EnðC Þ þ X 1 EnðAÞ EnðBÞ EnðAÞ EnðBÞ EnðBÞ EnðC Þ
ð13Þ
Relation between the strain component and the stress component can be given by a form as " ðσ σ Þ4 ðσ σ Þ3 ðσ σ Þ2 ε εy 4 X 1 max 4 þ X 2 max 3 þ X 3 max 2 ε εy ε εy ε εy # ðσ max σ Þ þ X5 ¼ 1 þ X4 ð14Þ ε εy
ε¼h En ¼
1 X 1 E4n þ X 2 E3n þX 3 E2n þ X 4 En þ X 5
σ max σ ; ε εy
i1=4 þ εy
σ ¼ σ max En ε εy ;
Enð0Þ ¼
σ max Δε
ð15Þ
It implies that we can use the variable En indirectly to obtain the relationship between the strain and the stress. Actually, in a numerical simulation, we may directly use an equivalent strainincrement (one of an experimental stress–strain relation) to calculate an incremental step of plastic deformation of material element. When an equivalent strain-increment is given, strainincrements involving other experimental stress–strain relations can be obtained based on a definition of equivalent hardening state [12–14]. Therefore, according to an increment of experimental strain component, we can directly calculate an associated increment of stress component as Δσ ¼
X 2 ðσ max σ Þ3 þ2X 3 ðσ max σ Þ2
ε εy þ 3X 4 ðσ max σ Þ ε εy 2 þ 4X 5 ε εy 3 2 3 Δε 4X 1 ðσ max σ Þ þ3X 2 ðσ max σ Þ ε εy þ2X 3 ðσ max σ Þ ε εy þ X 4 ε εy 3
2
ð16Þ To calculate the ratio of transverse to through thickness strainincrements, experimental strain-increment can be represented as Δε ¼
2 3 4X 1 ðσ max σ Þ3 þ3X 2 ðσ max σ Þ2 ε εy þ 2X 3 ðσ max σ Þ ε εy þ X 4 ε εy 2 3 Δσ 3 2 X 2 ðσ max σ Þ þ 2X 3 ðσ max σ Þ ε εy þ 3X 4 ðσ max σ Þ ε εy þ 4X 5 ε εy
ð17Þ Based on these coefficients, experimental stress–strain relations of rolled sheet metal displayed in Fig. 1 can be reproduced by this constitutive model as shown in Fig. 5. Experimental data of three middle points are ðεA ; σ A Þ ¼ ð0:0306; 343:8743Þ, ðεB ; σ B Þ ¼ ð0:0612; 410:7901Þ, ðεC ; σ C Þ ¼ ð0:1223; 481:9046Þ, ðεA ; σ A Þ ¼ ð 0:0148;
Fig. 6. Deviation analysis between this suggested forth order function and the power function. (a) Total fitting deviation dealing with experimental stress–strain relation in tensile direction. (b) Total fitting deviation dealing with experimental stress–strain relation in transverse direction.
343:4429Þ, ðεB ; σ B Þ ¼ ð 0:0294; 408:5908Þ, and ðεC ; σ C Þ ¼ ð 0:0591; 480:5016Þ. Since this constitutive model of experimental stress–strain relation takes more experimental data into account, predicting results can well fit the distribution of experimental data. Fig. 6 represents its deviation analysis. It can be seen that predicting accuracy of the forth order function is better than the second order function and much better than the power function. In relation to the prediction of strain-increment ratio (R-value), we can use Eq. (17) to calculate two strain-increments in stretching and transverse direction respectively. Except for the uniaxial tension, experimental stress–strain relations of various bi-axial tensions are also often used to characterize a constitutive equation for simulation of a forming process. Because of the increase of the usage of hydroforming technology in automotive industry, constitutive equations used for simulations of hydroforming processes would be better to be characterized by hydro-bulging data [15,16]. In order to understand more about these two suggested models in regression of experimental stress–strain relations regarding hydro-bulging tests, we discuss an additional case, in which experimental stress–strain relations have been performed through a hydrobulging test. In relation to this hydro-bulging test, ratio of axial to hoop stress components remains 5=8 through the controlling of inner pressure and axial force. This test responds to two experimental stress–strain relations as shown in Fig. 7. Based on these testing data, three different regressions of experimental stress–strain relations have been done. The first one is generated by the second order function, the second by the fourth order function and the third by the power function. Fig. 8 presents deviation analyses of these three models.
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Fig. 8. Deviation analysis within two suggested forth functions and a power function. (a) Deviations of three functions in hoop direction. (b) Deviations of three functions in axial direction.
behavior of rolled sheet metals. Actually, the phenomenon of changeable ratio of transverse to through thickness strain-increments has been pointed out a long time ago [17–20]. Many studies have attempted to consider this effect in associated models to describe material properties, such as the determination of forming limits [21], and the construction of constitutive model [4]. Recently, a phenomenological approach has been suggested to describe an evolution of R-value in a different trend [22], and the variation of the shape of plastic potential can also be represented by this model in reflection of the change of R-values. To understand a variation of the shape of plastic potential with the influence of experimental R-values, we may discuss it through a simple example. In plasticity, the gradient-based constitutive equations can be represented by a general form as
Fig. 7. Experimental stress–strain relations of hydro-bulging test reproduced by different functions. (a) Second order function. (b) Fourth order function. (c) Power function.
According to these predictions, it also displays that the suggested models respond to a higher accurate reproduction of experimental stress–strain relations than the power function.
3. Profile of the plastic potential with the influence of the regression of experimental data If two experimental stress–strain relations of uniaxial tension could be well formulated, changeable ratios (R-values) of transverse to through thickness strain-increments can then be taken into account in a suitable constitutive equation properly to describe the plastic flow
∂g dεpij ¼ dλ ∂σ ij
ð18Þ
This equation implies that the flow behavior of plastic strainincrements dεpij would be described totally by the plastic potential g. We may discuss a more general form of the plastic potential, i.e. the constitutive equation relates to a non-associated flow rule. It denotes that the plastic potential is just used to present the plastic flow behavior of material element, but not the yielding condition. Since plane stress states are more often adopted by a stamping simulation of rolled sheet metals, as an example in discussion, we may use the Hill's quadratic form [23] to define a plastic potential dealing with plane stress states as 1 þ R90 R0 2 R0 þ R90 1 þ 2R45 2 2R0 2 g ¼ σx þ σy σxσy þ τxy ¼ σ 20 1 þ R0 R90 1 þ R0 1 þ R0 R90 ð19Þ
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
7
where, R0 ; R90 ; R45 are the ratios of transverse to through thickness strain-increments dealing with a uniaxial tension in different direction to the rolling and they are all functions of hardening state, and σ 0 is the flow stress of uniaxial tension in rolling direction to describe an equivalent hardening state. Because these experimental data are used just for characterizing the feature of this plastic potential, predicting precision of the plastic potential would be seriously affected by the experimental data addressed in this model. Since profile of the plastic potential completely reflects the flow behavior of plastic strain-increments, now, we may discuss profile of the plastic potential influenced by the experimental strain ratios addressed in the plastic potential. We know that many constitutive models currently used in engineering applications adopt constant values of these strain ratios as regressions of experimental data. These ratios are determined by total strain values, but not strain-increments dealing with each deformation step. It means that profiles of associated plastic potentials are not varied during a strain-hardening process. However, according to the experimental data as shown in Fig. 4, variations of experimental strains in stretching and transverse directions cannot keep a constant ratio. This phenomenon means that deformation flow behaviors of rolled sheet metal element dealing with different stretching direction are not similar to each other, and they are also changed regarding different hardening state. To get an equivalent hardening state involving all of these tests, we can use the definition of dλ ¼ C [13,14] to obtain a relationship between uniaxial tensions in rolling and other directions. For example, when uniaxial tensions in rolling and transverse directions are related, we have 1 ∂g 1 dλ ¼ dεpij dσ ij dσ ij ¼ dεp0 dσ 0 ð2σ 0 dσ 0 Þ ∂σ ij " #1 2 1 þ R90 R0 ¼ dεp90 dσ 90 σ 90 dσ 90 ð20Þ 1 þ R0 R90 Based on this equivalent hardening state, we have a relation as 1 þ R90 R0 σ 90 Δεp90 ¼ Δεp0 ð21Þ 1 þR0 R90 σ 0
Fig. 9. Contours of plastic potentials relating to constant strain ratios and changeable strain ratios. (a) Subsequent plastic potential with constant strain ratios R0 ¼ R90 ¼ 2. (b) Subsequent plastic potential with variable strain ratios R90 o R0 . (c) Subsequent plastic potential with variable strain ratios R90 4R0 .
It means that when a strain-increment Δεp0 is given, an equivalent strain-increment Δεp90 can be determined together with Δεp0 to describe an equivalent hardening state of rolled sheet metals. According to this relation, we can obtain equivalent values of ratios R0 ; R90 to demonstrate profile of the plastic potential of Eq. (19) involving the plane principal stresses in rolling and transverse directions at each calculating hardening step. Fig. 9 shows some profiles of the plastic potential influenced by R values. Firstly, we discuss the cases that strain ratios R90 ;R0 remain constant values during a hardening process. If strain ratios R90 ;R0 remain constant values, all subsequent gradients of the plastic potential have the same value dealing with the same stress state, i.e. they all have a similar profile (see Fig. 9a). However, if strain ratios R90 ;R0 do not remain constant values during a strain-hardening progress, that is, strain ratios R90 ;R0 may change from a larger value to a smaller value, or reversely, subsequent gradients of the plastic potential are varied in relation to any of a hardening stage (see Fig. 9b and c). Fig. 9b presents the profiles of the plastic potential varied with changes of ratio R0 from a large value to a small value and ratio R90 from a small value to a large value. Fig. 9c shows a reverse result. Such variations of the profile of plastic potential mean that plastic flow behaviors of rolled sheet metals would be influenced by a hardening progress. Such deformation behavior of material element would considerably affect the spring-back prediction if elements of rolled sheet metals show strong anisotropy on its plastic flow process. Therefore, in order to obtain a reasonable spring-back
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Fig. 10. Experimental results of a rolled dual phase steel sheet. (a) Stress–strain relations in stretching and transverse directions. (b) Ratios of transverse to through thickness strains.
prediction regarding a stamped panel, taking variable strain ratios into account in the constitutive model is practically important. We need consider both of the anisotropic hardening behavior and the anisotropic plastic flow behavior of rolled sheet metals. Actually, ratios of transverse to through thickness strainincrements do not follow a certain rule to change their values. Different rolled sheet metals may respond to different demonstrations of these ratios. For example, Fig. 10 shows experimental results of a rolled dual phase steel sheet. These results demonstrate that all three uniaxial tensions have a similar tendency of the variation of strain ratios. That is, they are all changed from a large value to a small value in a tensile process. It denotes that a progressive hardening process would reduce thinning resistance of this sheet. However, according to the testing results of a 6A02 aluminum alloy sheet as shown in Fig. 11 we can see that variations of the strain ratio dealing with different loading direction are different. The uniaxial tension in rolling presents variation of strain ratio from a large value to a small value, and the uniaxial tensions in transverse and diagonal directions all demonstrate the tendency of strain ratio change from a small value to a large value. These experimental results mean that thinning
Fig. 11. Experimental results of an aluminum alloy plate. (a) Stress–strain relations in stretching and transverse directions. (b) Ratios of transverse to through thickness strains.
resistance of this aluminum alloy plate is reduced with a progressive hardening process in the rolling direction, and increased in the transverse and diagonal directions. According to these mechanical properties of rolled sheet metals, we can design a panel to fit a better position regarding blank setup.
4. Conclusions To take multi-experimental stress–strain relations into account in a constitutive equation for simulation of a sheet metal forming process has become a critical step to ensure its predicting precision. Because the existence of the anisotropy of rolled sheet metals, we need multi-experimental stress–strain relations obtained in different loading directions and different stress states to provide the information of anisotropic hardening and anisotropic plastic flow.
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
Therefore, a suitably constitutive equation needs to be able to address such multi-experimental stress–strain relations. Anisotropic plastic flow of rolled sheet metals is represented usually by the ratio of transverse to through thickness strains, i.e. two experimental stress–strain relations are involved. Commonly, this parameter is often given by a constant value in a constitutive model. However, many testing results have shown that it is a variable during a progressive hardening process. In order to obtain a suitable constitutive equation, regressions of these experimental stress– strain relations with a suitable constitutive model would become a challenge work because some empirical formulas currently used cannot treat all of these experimental stress–strain relations acceptably. To solve this issue, two general constitutive models are proposed. These two models can fit various experimental stress– strain relations of materials. Based on these models, experimental stress–strain relations dealing with complex stress states could be well reproduced, and a changeable ratio of transverse to through thickness strain-increments can also be reproduced properly to reflect the experimental data. These two models are very flexible, and they can also be used well to do the regression of experimental stress–strain relation dealing with hydro-bulging tests. Acknowledgments This work was supported by the National Natural Science Foundation of China (No.51405102) and the Fundamental Research Funds for the Central Universities (Grant no.HIT.NSRIF.2016093). The authors would like to take this opportunity to express their sincere appreciation to the funds. References [1] Mattiasson K, Sigvant M. An evaluation of some recent yield criteria for industrial simulations of sheet forming processes. Int J Mech Sci 2008;50 (4):774–87. [2] Hu WL. Characterized behaviors and corresponding yield criterion of anisotropic sheet metals. Mater Sci Eng A 2003;345:139–44.
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[3] Hu WL. An orthotropic yield criterion in a 3-D general stress state. Int J Plast 2005;21:1771–96. [4] Hu WL. Constitutive modeling of orthotropic sheet metals by presenting hardening-induced anisotropy. Int J Plast 2007;23:620–39. [5] Zang SL, Thuillier S, Le Port A, Manach PY. Prediction of anisotropy and hardening for metallic sheets in tension, simple shear and biaxial tension. Int J Mech Sci 2011;53(5):338–47. [6] Zhang SY, Leotoing L, Guines D, Thuillier S, Zang SL. Calibration of anisotropic yield criterion with conventional tests or biaxial test. Int J Mech Sci 2014;85:142–51. [7] Dafalias YF. The plastic spin. ASME J Appl Mech 1985;52:865–71. [8] Truong Qui HP, Lippmann H. Plastic spin and evolution of an anisotropic yield condition. Int J Mech Sci 2001;43(9):1969–83. [9] Yoshida F, Uemori T. A model of large-strain cyclic plasticity describing the Bauschinger effect and work hardening stagnation. Int J Plast 2002;18:661–86. [10] Gau JT, Kinzel GL. A new model for springback prediction in which the Bauschinger effect is considered. Int J Mech Sci 2001;43:1813–32. [11] Sanchez LR. Modeling of springback, strain rate and Bauschinger effects for two-dimensional steady state cyclic flow of sheet metal subjected to bending under tension. Int J Mech Sci 2010;52(3):429–39. [12] Hu WL. Equivalent strain-hardening work theorem. Philos Mag Lett 2004;84:7–14. [13] Hu WL, Wang ZR. Construction of a constitutive model in calculations of pressure-dependent material. Comput Mater Sci 2009;46:893–901. [14] Wang ZR, Hu WL, Hu L. Yield criteria and plastic stress–strain relations: theory and application. Beijing: Higher Education Press; 2014. [15] He ZB, Yuan SJ, Lin YL, Wang XS, Hu WL. Analytical model for tube hydrobulging test, part I: models for stress components and bulging zone profile. Int J Mech Sci 2014;87:297–306. [16] He ZB, Yuan SJ, Lin YL, Wang XS, Hu WL. Analytical model for tube hydrobulging tests, part II: linear model for pole thickness and its application. Int J Mech Sci 2014;87:307–15. [17] Hu H. The strain-dependence of plastic strain ratio (rm value) of deep drawing sheet steels determined by simple tension test. Metall Trans A 1975;6A:945–7. [18] Hu H. Effect of plastic strain on the r value of textured steel sheet. Metall Trans A 1975;6A:2307–9. [19] Arthey RP, Hutchinson WB. Variation of plastic strain ratio with strain level in steels. Metall Trans A 1981;12A:1817–22. [20] Wagoner RH. Plastic behavior of 70/30 brass sheet. Metall Trans A 1982;13A:1491–500. [21] Xu S, Weinmann KJ. Effect of deformation-dependent material parameters on forming limits of thin sheets. Int J Mech Sci 2000;42:677–92. [22] Safaei M, Lee MG, Zang SL, Waele WD. An evolutionary anisotropic model for sheet metals based on non-associated flow rule approach. Comput Mater Sci 2014;81:15–29. [23] Hill R. A theory of the yielding and plastic flow of anisotropic metals. Proc R Soc, Lond 1948;193A:281–97.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Constitutive models for regression of various experimental stress–strain relations Weilong Hu a, Yanli Lin b,n, Shijian Yuan c, Zhubin He c a
Troy Design & Manufacturing Co., 25111 Glendale, Redford, MI 48239, USA School of Materials Science & Engineering, Harbin Institute of Technology at Weihai, Weihai 264209, PR China c School of Materials Science & Engineering, Harbin Institute of Technology, Harbin 150001, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 2 March 2015 Received in revised form 16 June 2015 Accepted 10 July 2015 Available online 23 July 2015
Any of a theoretical constitutive model all needs to be characterized by proper experimental stress– strain relations to describe material behaviors in the domain of plastic deformation. It means that regression of experimental data by a suitable equation would be the most fundamental work. Some empirical equations currently used in engineering cannot always reproduce experimental stress–strain relations acceptably, particularly for experiments dealing with complex stress states and multiexperimental stress–strain relations. To improve the accuracy of numerical analysis, two new constitutive models for regression of various experimental stress–strain relations are suggested. One is simple relatively to another, but the complex one can give a more accurate reproduction of experimental stress– strain relation. Several uniaxial tensions of rolled sheet metals and a hydro-bulging test have been performed. Regressions of these experimental stress–strain relations have been done by two suggested models. Predicting precisions are discussed and compared with power functions. In relation to the uniaxial tension in different loading direction to the rolling, changeable ratios of transverse to through thickness strain-increments are explained, and their influence on the profile of plastic potential is presented. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Experimental stress–strain relation Constitutive model Regression Anisotropy Rolled sheet metal
1. Introduction Constructing an appropriate constitutive model is a critical step for obtaining a reasonable simulation result regarding a forming process. Experimental stress–strain relations dealing with different stress states are the most natural representation of material properties. Any of a constructed constitutive model all needs to be characterized by some experimental stress–strain relations. Capability of a constitutive model completely depends on the experimental data addressed in the model. Because of the rapid development of computer technology, numerical analyses have become the major method in engineering applications, and simulating accuracy has become the most important consideration in development of a new theory. Currently, many simplified models used to reproduce an experimental stress–strain relation, such as perfect rigid-plastic model and linear strain-hardening model etc., are seldom adopted in numerical analyses. A more accurate equation for reproducing experimental stress–strain relation is requested. If a real case could be well predicted by a constitutive model, the key issue would be that whether this model could
n
Corresponding author. Tel./fax: þ 86 631 5677408. E-mail address: [email protected] (Y. Lin).
http://dx.doi.org/10.1016/j.ijmecsci.2015.07.010 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
describe mechanical properties of materials good enough. It implies that multiple effects observed in different experiments could be accurately taken into account in the constitutive model. In relation to rolled sheet metals, experimental data include experimental stress–strain relations of uniaxial tension dealing with different loading direction, and they are used mainly to describe the anisotropic hardening of rolled sheet metals [1–6]. Some experimental data relate to loading path change that may vary the feature of original experimental stress–strain relations, such as the phenomena of plastic spin [7,8]. With respect to experimental stress– strain relations dealing with loading and reverted loading process, they are used to describe the Bauschinger effect [9–11]. Predicting accuracy of all constitutive models would basically rely on these original experimental stress–strain relations. It denotes that proper regressions of experimental stress–strain relations would be a more basic requirement to ensure that constitutive models could respond to an acceptable precision in predictions. If these original experimental stress–strain relations cannot be treated close to their realities, a proposed theory would significantly lose its function in description of a real deformation process. Because various materials present rather different mechanical properties, their stress–strain relations arising from different stress states cannot be defined all following a similar empirical formula, such as the most often used power law. Just dealing with a simple uniaxial
2
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
tension, if two experimental stress–strain relations of a rolled sheet metal all need to be used in describing a changeable ratio of transverse to through thickness strain-increments, these two experimental stress–strain relations cannot be always correctly reproduced by two power expressions. However, these two experimental stress–strain relations are very important factors used to present the anisotropic flow behavior of rolled sheet metals. To solve this problem, we may directly use the experimental data in a theoretical constitutive model. However, if several experimental stress–strain relations all need to be addressed in this constitutive model to describe anisotropic hardening behavior of rolled sheet metals, equivalent increments of stress and strain components dealing with different experiments are also hard to be defined. If we insist on using an empirical formula, we may take a risk of some degree of the deviation between the experimental stress– strain relation and the reproduced stress–strain relation. For some cases, such deviation is actually unacceptable, particularly for the case that analyzing results are very sensitive to the hardening behavior of experimental stress–strain relation, for example the spring-back simulation of a stamped panel. No matter how many effects are introduced in a constitutive model, original experimental stress– strain relations would always be the first scale to affect its predicting accuracy. In order to obtain a more accurate reproduction of experimental stress–strain relation, two new formulations are introduced in this paper. They can be used as constitutive models for regression of various experimental stress–strain relations. Based on these models, a formulaic experimental stress–strain relation can be easily obtained to fit an acceptable accuracy. We can base on three defined points of experimental data, for example the initial yield, the maximum stress point and a point between the initial yield and the maximum stress point, to generate a constitutive model of experimental stress–strain relation. If we would like to obtain a more accurate reproduction of experimental stress–strain relation, we can also use five defined points regarding the experimental data to generate a formula of experimental stress–strain relation. By using these models, many benefits could be obtained. We can directly determine a formula of experimental stress–strain relation and do not need to clean up the serrate shape of the original testing data. It would save lots work. No matter what a stress state of the stress–strain relation is related, they all can be treated by the same constitutive model and meet an accurate requirement. Several experimental stress–strain relations are discussed to show the use of these proposed constitutive models.
2. Constitutive models of experimental stress–strain relation In order to make regression of experimental stress–strain relation more close to the reality, we attempt to propose a new constitutive model. With respect to a uniaxial tension, it deals with two experimental stress–strain relations. When doing regressions of these two experimental stress–strain relations, this model does not need all original experimental data. That is, only several points on the experimental stress–strain relation are selected to do its regression. These data can be the initial yield point, the maximum stress point and some points between the initial yield and the maximum stress, for example 5%, 10%, 15%, or 20% strains and so on. 2.1. Constitutive model with a second order function To define such a constitutive model to match an experimental stress–strain relation, at least, we need three points to generate a curve as a reproduction of this experimental stress–strain relation. We may select the initial yield point, the maximum stress point and a point A between these two points, such as the 5% strain point, or some other point with different strain value. It means
that a suitable function would have three pending coefficients to address these three experimental points. We may propose a function as a form 2 X 1 ðσ max σ Þ2 þ X 2 ε εy ðσ max σ Þ þ X 3 ε εy 1 ¼ 0 ð1Þ where σ max is the maximum stress, εy is the initial yield strain, and X 1 ; X 2 ; X 3 are the pending coefficients to address the experimental data. Therefore, these three coefficients can be obtained as 1
X1 ¼
;
X3 ¼
1
σ max σ y 2 εmax εy 2 " 2 # εA εy σ max σ A 2 1 X2 ¼ 1 εmax εy σ max σ y ðσ max σ A Þ εA εy
ð2Þ
where σ y ; εmax are the initial yield stress and the maximum strain, and σ A ; εA are the stress and strain dealing with the point A. Therefore, Eq. (1) can be finally represented as " # σÞ εA εy 2 ðσ max σ Þ2 σ max σ A 2 ε εy ðσ max þ 1 εmax εy σ max σ y ðσ max σ A Þ εA εy σ max σ y 2
ε εy 2 1 ¼ 0 εmax εy 2
ð3Þ
þ
Since this is a second order function, we can directly obtain the relationship between the stress and the strain as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i X2 X 22 1h σ ¼ σ max þ ε εy ε εy 2 X 3 ε εy 2 1 ð4Þ 2 X 2X 1 1 4X 1
ε ¼ εy
X2 ðσ max σ Þ þ 2X 3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i X 22 1h ðσ max σ Þ2 X 1 ðσ max σ Þ2 1 2 X3 4X 3 ð5Þ
We can also use another method to obtain the correlation between the stress and the strain, i.e. Eq. (1) can be given by a form as 2 ε εy 2 X 1 ðσ max σ2Þ þ X 2 ðσðεmax εy σÞ Þ þ X 3 ¼ 1 ðε εy Þ
ε εy ¼
1 X 1 E2n þX 2 En þ X 3
1=2
σ max σ ; σ max σ ¼ En ε εy ε εy σ ¼ σ max En ε εy En ¼
ð6Þ
Since we more often use a numerical method to analyze a real process, based on Eq. (1), we can get a relationship between the stress increment and the strain increment as X ðσ σ Þ þ 2X 3 ε εy Δε Δσ ¼ 2 max ð7Þ 2X 1 ðσ max σ Þ þX 2 ε εy Actually, we can also base on Eqs. (4) and (5) to obtain an incremental relationship between stress and strain. According to this constitutive model, we can get reproductions of these experimental stress–strain relations. Since this constitutive model only requires three points to determine the profile of experimental stress–strain relation, we may vary the point A to get a better fit of experimental data. For this case, strain and stress values of three selected points are 0.0019 ( 0.000867), 224.178 as the initial yield point, 0.0612 ( 0.0294), 410.7901 as the middle point, and 0.24461 ( 0.11817), 547.9195 as the maximum stress point, where values in the brackets mean the strains in transverse direction. We can use these data to determine three coefficients in the constitutive model of Eq. (1). Finally, two stress–strain relations predicted by this constitutive model are shown in Fig. 1. In relation to the stress–strain relation in transverse direction, we still use a
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
3
base on Eq. (5) to obtain relations of strain-increment as 2 3
2 4X X X ð σ σ Þ max 3 1 6 7 1 2 7 Δεt ¼ 6 h i52X Δσ 4X 2 þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 2 X 2 ðσ max σ Þ 4X 3 X 1 ðσ max σ Þ 1 2
3
2 ^ ^ ^ 4 X X ð σ σ Þ X max 6 3 1 7 1 2 Δεw ¼ 6X^ 2 þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 4 h i5 ^ Δσ 2 2X 3 2 2 X^ 2 ðσ max σ Þ 4X^ 3 X^ 1 ðσ max σ Þ 1
ð9Þ
Fig. 1. Experimental stress–strain relations reproduced constitutive model.
by the suggested
where, X^ 1 ; X^ 2 ; X^ 3 are coefficients determined by the experimental strain and stress values in transverse direction, and all strain values used are positive, but not negative values. Substituting relations of Eq. (9) into Eq. (8), we can get the ratio of transverse to through thickness strain-increments. Fig. 4 displays the distribution of this ratio varying with strain value in tensile direction. To compare it with the result obtained based on two power functions, we have
Δεt ¼
1 1=n
nk
σ tð1 nÞ=n ;
Δεw ¼
1 n^ k^
1=n^
σ ðt
1 n^ Þ=n^
1=n
R¼
Fig. 2. Experimental stress–strain relations reproduced by two power functions.
positive strain value to determine its stress value, i.e. we do not need to change the representation of this constitutive model. From this picture, predictions seem well to fit the distributions of experimental data. In order to evaluate its predicting accuracy, we may compare it with the prediction of a power function. Fig. 2 shows two stress–strain relations predicted by two power functions. Table 1 gives a summary of these two power functions dealing with the regressions of experimental stress–strain relations. Fig. 3 presents deviation analysis. Based on this comparison, predicting accuracy of this suggested function looks much better than the power function. After we get constitutive models of experimental stress–strain relations dealing with both stretching and transverse directions, we can now obtain a relation of anisotropic parameter (R-value) distributed in different hardening state. Because of the existence of the anisotropy, this parameter cannot be a constant value. Capability of representing this behavior of rolled sheet metals would give much benefit on a theoretical constitutive model for simulating a stamping process, including the spring-back prediction. Suppose that material element is an uncompressible body, we have R¼
Δεw Δεw 1 1 ¼ ¼ ¼ 1 þ Δεt =Δεw Δεt = Δεw 1 Δεz Δεw þ Δεt
ð8Þ
where Δεt ; Δεw ; Δεz are the strain-increments in stretching, transverse and through thickness directions respectively. We may
1 nk ð1 nÞ=n ð1 n^ Þ=n^ 1=n σ½ nk ¼ Δεt = Δεw 1 n^ k^ 1=n^ t
ð10Þ
^ n^ are material parameters involving the power where, k; n; k; functions. Two calculation results are also shown in Fig. 4. According to this result we can see that the ratio of transverse to through thickness strain-increments is varied with a progressive hardening process. It means that if we use a constant value of the ratio in a constitutive model, like the use currently in stamping simulations, it should affect the predicting accuracy, particularly for the spring-back prediction. One thing that needs to be paid attention is that the testing data of Rvalue expressed by strain-increments are very unstable. It means that regression precision of R-value would completely depend on two regression functions of experimental stress–strain relations. We cannot obtain an acceptable regression of R-value directly from testing data. The testing data of R-value shown in Fig. 4 were measured by each calculation step around 3% tensile strain-increments. Generally, this model can well reproduce experimental stress– strain relation of rolled sheet metals dealing with any of a testing stress states. However, if we would like to obtain a more accurate regression of experimental stress–strain relation, we may use another constitutive model suggested as follows. 2.2. Constitutive model with a fourth order function To obtain a higher accurate reproduction of experimental stress–strain relation, we may need a constitutive model that could take more experimental data into account to improve its predicting precision. With respect to this purpose, we now propose a new constitutive model for regression of experimental stress–strain relations, i.e. 2 X 1 ðσ max σ Þ4 þ X 2 ðσ max σ Þ3 ε εy þX 3 ðσ max σ Þ2 ε εy 3 4 þ X 4 ðσ max σ Þ ε εy þ X 5 ε εy ¼ 1 ð11Þ This is a fourth order function. Since this function has five pending coefficients, it can address more experimental data finely to define its characteristic in prediction of experimental stress– strain relations. We can select five points from the data of experimental stress–strain relation to characterize this function. They are the initial yield point, maximum stress point and three points A; B; C between the initial and maximum stress points.
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Table 1 Summary of two power functions. (a) Regression of a power expression in tensile direction K
n
Value 754.76731
D
Standard error 1.45603
DF Regression 2 Residual 1531 Uncorrected total 1533 Corrected total 1532 (b) Regression of a power expression in transverse direction D
K
F F
Statistics
Value 0.22019
Standard error 8.57106E 4
Reduced Chi-Square 160.65067
Adj. R-Square 0.98861
Sum of squares 3.11417E8 245956.16937 3.11663E8 2.16043E7
Mean square 1.55708E8 160.65067
F value 969235.55591
Prob4F 0
n
Statistics
Value 907.64839
Standard error 3.26731
Value 0.22978
Standard error 0.00124
Reduced Chi-Square 306.65415
Adj. R-Square 0.97726
Regression Residual Uncorrected total Corrected total
DF 2 1465 1467 1466
Sum of squares 3.01417E8 449248.32331 3.01866E8 1.97654E7
Mean square 1.50708E8 306.65415
F value 491460.63464
Prob4F 0
Fig. 4. Changeable ratio of transverse to through thickness strain-increments.
stresses of a stress–strain relation between the initial and maximum stress points. Substituting stress and strain data of these five points into Eq. (11), we finally obtain X1 ¼
1
σ max σ y
Fig. 3. Deviation analysis between the suggested function and the power function. (a) Total fitting deviation dealing with experimental stress–strain relation in tensile direction. (b) Total fitting deviation dealing with experimental stress–strain relation in transverse direction.
Suppose that these yielding stresses and strains are represented by EnðAÞ ¼
σ max σ A ; εA εy
EnðBÞ ¼
σ max σ B ; εB εy
EnðC Þ ¼
σ max σ C εC εy
ð12Þ
where, εA ; εB ; εC ; σ A ; σ B ; σ C are the experimental strains and
4 ;
X5 ¼
1
εmax εy
4
EnðAÞ 1 1 1 X4 2 þ X 5 3 X 1 EnðAÞ EnðAÞ EnðAÞ ðσ max σ A Þ4 EnðAÞ " ! EnðAÞ EnðBÞ 1 1 X3 ¼ þ X 5 ðσ max σ B Þ4 ðσ max σ A Þ4 E3nðAÞ E3nðBÞ EnðAÞ EnðBÞ 1 1 X 4 þ þX 1 EnðAÞ EnðBÞ EnðBÞ EnðAÞ EnðAÞ EnðBÞ " ! EnðAÞ EnðC Þ 1 1 þ X X4 ¼ 5 ðσ max σ C Þ4 ðσ max σ A Þ4 E3nðAÞ E3nðC Þ X2 ¼ X3
W. Hu et al. / International Journal of Mechanical Sciences 101-102 (2015) 1–9
5
Fig. 5. Fine reproduction of experimental stress–strain relation by a more accurate model.
EnðAÞ EnðBÞ E2nðC Þ þ X 1 EnðAÞ EnðC Þ EnðAÞ EnðC Þ EnðBÞ EnðC Þ " ! EnðAÞ EnðBÞ 1 1 þ X5 3 3 ðσ max σ B Þ4 ðσ max σ A Þ4 EnðAÞ EnðBÞ EnðAÞ E2nðBÞ EnðC Þ þ X 1 EnðAÞ EnðBÞ EnðAÞ EnðBÞ EnðBÞ EnðC Þ
ð13Þ
Relation between the strain component and the stress component can be given by a form as " ðσ σ Þ4 ðσ σ Þ3 ðσ σ Þ2 ε εy 4 X 1 max 4 þ X 2 max 3 þ X 3 max 2 ε εy ε εy ε εy # ðσ max σ Þ þ X5 ¼ 1 þ X4 ð14Þ ε εy
ε¼h En ¼
1 X 1 E4n þ X 2 E3n þX 3 E2n þ X 4 En þ X 5
σ max σ ; ε εy
i1=4 þ εy
σ ¼ σ max En ε εy ;
Enð0Þ ¼
σ max Δε
ð15Þ
It implies that we can use the variable En indirectly to obtain the relationship between the strain and the stress. Actually, in a numerical simulation, we may directly use an equivalent strainincrement (one of an experimental stress–strain relation) to calculate an incremental step of plastic deformation of material element. When an equivalent strain-increment is given, strainincrements involving other experimental stress–strain relations can be obtained based on a definition of equivalent hardening state [12–14]. Therefore, according to an increment of experimental strain component, we can directly calculate an associated increment of stress component as Δσ ¼
X 2 ðσ max σ Þ3 þ2X 3 ðσ max σ Þ2
ε εy þ 3X 4 ðσ max σ Þ ε εy 2 þ 4X 5 ε εy 3 2 3 Δε 4X 1 ðσ max σ Þ þ3X 2 ðσ max σ Þ ε εy þ2X 3 ðσ max σ Þ ε εy þ X 4 ε εy 3
2
ð16Þ To calculate the ratio of transverse to through thickness strainincrements, experimental strain-increment can be represented as Δε ¼
2 3 4X 1 ðσ max σ Þ3 þ3X 2 ðσ max σ Þ2 ε εy þ 2X 3 ðσ max σ Þ ε εy þ X 4 ε εy 2 3 Δσ 3 2 X 2 ðσ max σ Þ þ 2X 3 ðσ max σ Þ ε εy þ 3X 4 ðσ max σ Þ ε εy þ 4X 5 ε εy
ð17Þ Based on these coefficients, experimental stress–strain relations of rolled sheet metal displayed in Fig. 1 can be reproduced by this constitutive model as shown in Fig. 5. Experimental data of three middle points are ðεA ; σ A Þ ¼ ð0:0306; 343:8743Þ, ðεB ; σ B Þ ¼ ð0:0612; 410:7901Þ, ðεC ; σ C Þ ¼ ð0:1223; 481:9046Þ, ðεA ; σ A Þ ¼ ð 0:0148;
Fig. 6. Deviation analysis between this suggested forth order function and the power function. (a) Total fitting deviation dealing with experimental stress–strain relation in tensile direction. (b) Total fitting deviation dealing with experimental stress–strain relation in transverse direction.
343:4429Þ, ðεB ; σ B Þ ¼ ð 0:0294; 408:5908Þ, and ðεC ; σ C Þ ¼ ð 0:0591; 480:5016Þ. Since this constitutive model of experimental stress–strain relation takes more experimental data into account, predicting results can well fit the distribution of experimental data. Fig. 6 represents its deviation analysis. It can be seen that predicting accuracy of the forth order function is better than the second order function and much better than the power function. In relation to the prediction of strain-increment ratio (R-value), we can use Eq. (17) to calculate two strain-increments in stretching and transverse direction respectively. Except for the uniaxial tension, experimental stress–strain relations of various bi-axial tensions are also often used to characterize a constitutive equation for simulation of a forming process. Because of the increase of the usage of hydroforming technology in automotive industry, constitutive equations used for simulations of hydroforming processes would be better to be characterized by hydro-bulging data [15,16]. In order to understand more about these two suggested models in regression of experimental stress–strain relations regarding hydro-bulging tests, we discuss an additional case, in which experimental stress–strain relations have been performed through a hydrobulging test. In relation to this hydro-bulging test, ratio of axial to hoop stress components remains 5=8 through the controlling of inner pressure and axial force. This test responds to two experimental stress–strain relations as shown in Fig. 7. Based on these testing data, three different regressions of experimental stress–strain relations have been done. The first one is generated by the second order function, the second by the fourth order function and the third by the power function. Fig. 8 presents deviation analyses of these three models.
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Fig. 8. Deviation analysis within two suggested forth functions and a power function. (a) Deviations of three functions in hoop direction. (b) Deviations of three functions in axial direction.
behavior of rolled sheet metals. Actually, the phenomenon of changeable ratio of transverse to through thickness strain-increments has been pointed out a long time ago [17–20]. Many studies have attempted to consider this effect in associated models to describe material properties, such as the determination of forming limits [21], and the construction of constitutive model [4]. Recently, a phenomenological approach has been suggested to describe an evolution of R-value in a different trend [22], and the variation of the shape of plastic potential can also be represented by this model in reflection of the change of R-values. To understand a variation of the shape of plastic potential with the influence of experimental R-values, we may discuss it through a simple example. In plasticity, the gradient-based constitutive equations can be represented by a general form as
Fig. 7. Experimental stress–strain relations of hydro-bulging test reproduced by different functions. (a) Second order function. (b) Fourth order function. (c) Power function.
According to these predictions, it also displays that the suggested models respond to a higher accurate reproduction of experimental stress–strain relations than the power function.
3. Profile of the plastic potential with the influence of the regression of experimental data If two experimental stress–strain relations of uniaxial tension could be well formulated, changeable ratios (R-values) of transverse to through thickness strain-increments can then be taken into account in a suitable constitutive equation properly to describe the plastic flow
∂g dεpij ¼ dλ ∂σ ij
ð18Þ
This equation implies that the flow behavior of plastic strainincrements dεpij would be described totally by the plastic potential g. We may discuss a more general form of the plastic potential, i.e. the constitutive equation relates to a non-associated flow rule. It denotes that the plastic potential is just used to present the plastic flow behavior of material element, but not the yielding condition. Since plane stress states are more often adopted by a stamping simulation of rolled sheet metals, as an example in discussion, we may use the Hill's quadratic form [23] to define a plastic potential dealing with plane stress states as 1 þ R90 R0 2 R0 þ R90 1 þ 2R45 2 2R0 2 g ¼ σx þ σy σxσy þ τxy ¼ σ 20 1 þ R0 R90 1 þ R0 1 þ R0 R90 ð19Þ
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where, R0 ; R90 ; R45 are the ratios of transverse to through thickness strain-increments dealing with a uniaxial tension in different direction to the rolling and they are all functions of hardening state, and σ 0 is the flow stress of uniaxial tension in rolling direction to describe an equivalent hardening state. Because these experimental data are used just for characterizing the feature of this plastic potential, predicting precision of the plastic potential would be seriously affected by the experimental data addressed in this model. Since profile of the plastic potential completely reflects the flow behavior of plastic strain-increments, now, we may discuss profile of the plastic potential influenced by the experimental strain ratios addressed in the plastic potential. We know that many constitutive models currently used in engineering applications adopt constant values of these strain ratios as regressions of experimental data. These ratios are determined by total strain values, but not strain-increments dealing with each deformation step. It means that profiles of associated plastic potentials are not varied during a strain-hardening process. However, according to the experimental data as shown in Fig. 4, variations of experimental strains in stretching and transverse directions cannot keep a constant ratio. This phenomenon means that deformation flow behaviors of rolled sheet metal element dealing with different stretching direction are not similar to each other, and they are also changed regarding different hardening state. To get an equivalent hardening state involving all of these tests, we can use the definition of dλ ¼ C [13,14] to obtain a relationship between uniaxial tensions in rolling and other directions. For example, when uniaxial tensions in rolling and transverse directions are related, we have 1 ∂g 1 dλ ¼ dεpij dσ ij dσ ij ¼ dεp0 dσ 0 ð2σ 0 dσ 0 Þ ∂σ ij " #1 2 1 þ R90 R0 ¼ dεp90 dσ 90 σ 90 dσ 90 ð20Þ 1 þ R0 R90 Based on this equivalent hardening state, we have a relation as 1 þ R90 R0 σ 90 Δεp90 ¼ Δεp0 ð21Þ 1 þR0 R90 σ 0
Fig. 9. Contours of plastic potentials relating to constant strain ratios and changeable strain ratios. (a) Subsequent plastic potential with constant strain ratios R0 ¼ R90 ¼ 2. (b) Subsequent plastic potential with variable strain ratios R90 o R0 . (c) Subsequent plastic potential with variable strain ratios R90 4R0 .
It means that when a strain-increment Δεp0 is given, an equivalent strain-increment Δεp90 can be determined together with Δεp0 to describe an equivalent hardening state of rolled sheet metals. According to this relation, we can obtain equivalent values of ratios R0 ; R90 to demonstrate profile of the plastic potential of Eq. (19) involving the plane principal stresses in rolling and transverse directions at each calculating hardening step. Fig. 9 shows some profiles of the plastic potential influenced by R values. Firstly, we discuss the cases that strain ratios R90 ;R0 remain constant values during a hardening process. If strain ratios R90 ;R0 remain constant values, all subsequent gradients of the plastic potential have the same value dealing with the same stress state, i.e. they all have a similar profile (see Fig. 9a). However, if strain ratios R90 ;R0 do not remain constant values during a strain-hardening progress, that is, strain ratios R90 ;R0 may change from a larger value to a smaller value, or reversely, subsequent gradients of the plastic potential are varied in relation to any of a hardening stage (see Fig. 9b and c). Fig. 9b presents the profiles of the plastic potential varied with changes of ratio R0 from a large value to a small value and ratio R90 from a small value to a large value. Fig. 9c shows a reverse result. Such variations of the profile of plastic potential mean that plastic flow behaviors of rolled sheet metals would be influenced by a hardening progress. Such deformation behavior of material element would considerably affect the spring-back prediction if elements of rolled sheet metals show strong anisotropy on its plastic flow process. Therefore, in order to obtain a reasonable spring-back
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Fig. 10. Experimental results of a rolled dual phase steel sheet. (a) Stress–strain relations in stretching and transverse directions. (b) Ratios of transverse to through thickness strains.
prediction regarding a stamped panel, taking variable strain ratios into account in the constitutive model is practically important. We need consider both of the anisotropic hardening behavior and the anisotropic plastic flow behavior of rolled sheet metals. Actually, ratios of transverse to through thickness strainincrements do not follow a certain rule to change their values. Different rolled sheet metals may respond to different demonstrations of these ratios. For example, Fig. 10 shows experimental results of a rolled dual phase steel sheet. These results demonstrate that all three uniaxial tensions have a similar tendency of the variation of strain ratios. That is, they are all changed from a large value to a small value in a tensile process. It denotes that a progressive hardening process would reduce thinning resistance of this sheet. However, according to the testing results of a 6A02 aluminum alloy sheet as shown in Fig. 11 we can see that variations of the strain ratio dealing with different loading direction are different. The uniaxial tension in rolling presents variation of strain ratio from a large value to a small value, and the uniaxial tensions in transverse and diagonal directions all demonstrate the tendency of strain ratio change from a small value to a large value. These experimental results mean that thinning
Fig. 11. Experimental results of an aluminum alloy plate. (a) Stress–strain relations in stretching and transverse directions. (b) Ratios of transverse to through thickness strains.
resistance of this aluminum alloy plate is reduced with a progressive hardening process in the rolling direction, and increased in the transverse and diagonal directions. According to these mechanical properties of rolled sheet metals, we can design a panel to fit a better position regarding blank setup.
4. Conclusions To take multi-experimental stress–strain relations into account in a constitutive equation for simulation of a sheet metal forming process has become a critical step to ensure its predicting precision. Because the existence of the anisotropy of rolled sheet metals, we need multi-experimental stress–strain relations obtained in different loading directions and different stress states to provide the information of anisotropic hardening and anisotropic plastic flow.
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Therefore, a suitably constitutive equation needs to be able to address such multi-experimental stress–strain relations. Anisotropic plastic flow of rolled sheet metals is represented usually by the ratio of transverse to through thickness strains, i.e. two experimental stress–strain relations are involved. Commonly, this parameter is often given by a constant value in a constitutive model. However, many testing results have shown that it is a variable during a progressive hardening process. In order to obtain a suitable constitutive equation, regressions of these experimental stress– strain relations with a suitable constitutive model would become a challenge work because some empirical formulas currently used cannot treat all of these experimental stress–strain relations acceptably. To solve this issue, two general constitutive models are proposed. These two models can fit various experimental stress– strain relations of materials. Based on these models, experimental stress–strain relations dealing with complex stress states could be well reproduced, and a changeable ratio of transverse to through thickness strain-increments can also be reproduced properly to reflect the experimental data. These two models are very flexible, and they can also be used well to do the regression of experimental stress–strain relation dealing with hydro-bulging tests. Acknowledgments This work was supported by the National Natural Science Foundation of China (No.51405102) and the Fundamental Research Funds for the Central Universities (Grant no.HIT.NSRIF.2016093). The authors would like to take this opportunity to express their sincere appreciation to the funds. References [1] Mattiasson K, Sigvant M. An evaluation of some recent yield criteria for industrial simulations of sheet forming processes. Int J Mech Sci 2008;50 (4):774–87. [2] Hu WL. Characterized behaviors and corresponding yield criterion of anisotropic sheet metals. Mater Sci Eng A 2003;345:139–44.
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[3] Hu WL. An orthotropic yield criterion in a 3-D general stress state. Int J Plast 2005;21:1771–96. [4] Hu WL. Constitutive modeling of orthotropic sheet metals by presenting hardening-induced anisotropy. Int J Plast 2007;23:620–39. [5] Zang SL, Thuillier S, Le Port A, Manach PY. Prediction of anisotropy and hardening for metallic sheets in tension, simple shear and biaxial tension. Int J Mech Sci 2011;53(5):338–47. [6] Zhang SY, Leotoing L, Guines D, Thuillier S, Zang SL. Calibration of anisotropic yield criterion with conventional tests or biaxial test. Int J Mech Sci 2014;85:142–51. [7] Dafalias YF. The plastic spin. ASME J Appl Mech 1985;52:865–71. [8] Truong Qui HP, Lippmann H. Plastic spin and evolution of an anisotropic yield condition. Int J Mech Sci 2001;43(9):1969–83. [9] Yoshida F, Uemori T. A model of large-strain cyclic plasticity describing the Bauschinger effect and work hardening stagnation. Int J Plast 2002;18:661–86. [10] Gau JT, Kinzel GL. A new model for springback prediction in which the Bauschinger effect is considered. Int J Mech Sci 2001;43:1813–32. [11] Sanchez LR. Modeling of springback, strain rate and Bauschinger effects for two-dimensional steady state cyclic flow of sheet metal subjected to bending under tension. Int J Mech Sci 2010;52(3):429–39. [12] Hu WL. Equivalent strain-hardening work theorem. Philos Mag Lett 2004;84:7–14. [13] Hu WL, Wang ZR. Construction of a constitutive model in calculations of pressure-dependent material. Comput Mater Sci 2009;46:893–901. [14] Wang ZR, Hu WL, Hu L. Yield criteria and plastic stress–strain relations: theory and application. Beijing: Higher Education Press; 2014. [15] He ZB, Yuan SJ, Lin YL, Wang XS, Hu WL. Analytical model for tube hydrobulging test, part I: models for stress components and bulging zone profile. Int J Mech Sci 2014;87:297–306. [16] He ZB, Yuan SJ, Lin YL, Wang XS, Hu WL. Analytical model for tube hydrobulging tests, part II: linear model for pole thickness and its application. Int J Mech Sci 2014;87:307–15. [17] Hu H. The strain-dependence of plastic strain ratio (rm value) of deep drawing sheet steels determined by simple tension test. Metall Trans A 1975;6A:945–7. [18] Hu H. Effect of plastic strain on the r value of textured steel sheet. Metall Trans A 1975;6A:2307–9. [19] Arthey RP, Hutchinson WB. Variation of plastic strain ratio with strain level in steels. Metall Trans A 1981;12A:1817–22. [20] Wagoner RH. Plastic behavior of 70/30 brass sheet. Metall Trans A 1982;13A:1491–500. [21] Xu S, Weinmann KJ. Effect of deformation-dependent material parameters on forming limits of thin sheets. Int J Mech Sci 2000;42:677–92. [22] Safaei M, Lee MG, Zang SL, Waele WD. An evolutionary anisotropic model for sheet metals based on non-associated flow rule approach. Comput Mater Sci 2014;81:15–29. [23] Hill R. A theory of the yielding and plastic flow of anisotropic metals. Proc R Soc, Lond 1948;193A:281–97.