Ceramics International xxx (2018) xxx-xxx

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Ceramics International

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journal homepage: www.elsevier.com

Methodology for dependence-based integrated constitutive modelling: An illustrative application to SiCp /Al composites Junfeng Xianga , Lijing Xiea , b , ⁎ , Feinong Gaoa , Jie Yia , Siqin Panga , b , Xibin Wanga , b a b

School of Mechanical Engineering, Beijing Institute of Technology, 100081 Beijing, China Key Laboratory of Advanced Machining, Beijing Institute of Technology, 100081 Beijing, China

ABSTRACT

Keywords: Constitutive modelling Multi-objective Parameter identification SiCp /Al composites

In industrial forming and machining process, the large plastic deformation of material takes place in wide loading ranges of strain-rate and forming temperature. A satisfactory modelling of quasi-static and dynamic material behaviors is of great importance for understanding physical process and processes optimization. A dependence-based integrated methodology, together with an improved weighted multi-objective parameter identification strategy is presented for the development of phenomenological constitutive model and the parameter identification using experimental data from quasi-static and dynamic tests with instantaneous strain rate variations and plastic strain-related temperature changes. The improved multi-objective parameter identification model is reformulated by introducing three weighting factors for valuing different measure errors and fit standard errors in individual objective function corresponding to each test, considering the sampling point number and active material parameter number under different loading conditions, and balancing optimization opportunity of quasi-static and dynamic sub-objective functions. The methodology is verified for feasibility through illustrative constitutive identification for SiCp /Al composites. This may provide a methodology of constitutive modelling for predicting material behaviors in quasi-static and dynamic modes equally well.

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ARTICLE INFO

1. Introduction

In manufacturing industry, almost all material forming and machining processes consist of deforming plastically workpiece under the forces applied by a die or tool [1]. Understanding of material flow behavior under different deformation modes is gaining critical importance in planning of material forming and manufacturing processes, selection of die/tool, design feasibility of final products [2–4]. A schematic of the dependence of analytical modelling and finite element modelling on constitutive model in material forming and machining processes is illustrated in Fig. 1. Therefore, it is necessary to develop/establish constitutive model reflecting the yield criteria, hardening laws and flow rule of the material under com

⁎

bined loading conditions of deformation rate and forming temperature. The accuracy of numerical simulation available for industrial applications depends not only upon the applicability and flexibility of material constitutive model, but also the scheme applied to determine material parameters [5]. With the continuous emergence of new materials, classical constitutive models (e.g., Johnson-Cook [6], Arrhenius-type [7], Zerilli-Armstrong [8], etc.), much less so-called unified constitutive models that tend to describe a given crystalline structure or crystalline system, cannot accurately predict the flow behaviors of those new materials in a wide loading range. These pose encountering challenges to the determination of material constitutive models. For improving the applicability and flexibility of the above classical constitutive models, it is necessary to develop or mod

Corresponding author at: School of Mechanical Engineering, Beijing Institute of Technology, 100081 Beijing, China. Email address: [email protected] (L. Xie)

https://doi.org/10.1016/j.ceramint.2018.03.257 Received 21 December 2017; Received in revised form 15 February 2018; Accepted 28 March 2018 Available online xxx 0272-8842/ © 2018.

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Fig. 1. Incorporation of constitutive model in analytical modelling and finite element modelling of material forming and machining processes.

ify advanced constitutive models based on the classical phenomenological constitutive models by introducing the coupled effects (e.g., between strain-rate and temperatures [9], between strain and temperature [10], between other physical or mechanical properties and strain [11,12], and among strain, strain-rate and temperature [13]), some special mechanical behaviors (e.g., strain-softening at high strain [14], scale effect [15], ratcheting effect [16], anisotropy [17], plastic strain gradient [18], etc.), and other experimentally observed physical or mechanical properties varying with deformation such as density in foam metals [19], microscopic damage in composites [20], phase transformation [21], and grain evolution [8,22], etc. This would also lead to the complexity of constitutive model involving more material parameters, and increase the experimental cost for developing constitutive model and identifying its material parameters, of which some can be attained through microscopic observation [20], and some can be also fit

to numerous experimental data in quasi-static and dynamic modes [9–12]. Therefore, the major difficulties for widespread application of phenomenological constitutive model for industrial simulation consist in a larger number of mechanical tests over a wide range of loading conditions needed for using classical identification methodology of constitutive model, identification of coupled relations among explanatory variables, and the ensuing hard identification job of material parameters involved in constitutive characterization [23,24]. The aim of this paper is to present an improved methodology for dependence-based integrated constitutive modelling, together with multi-objective material parameters identification strategy. Using an illustrative, yet complex example for SiC particulate reinforced aluminum matrix composites using experimental data from quasi-static and dynamic tests, the methodology is verified for feasibility by comparing the overall fit quality of the identified constitutive model to experimental data.

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A phenomenological constitutive model is actually an empirical relationship for mechanical characterization of complex material responses in different loading regimes [25]. The general formalism of a one-dimensional phenomenological model is to represent the dependence of flow stress on strain, strain-rate, temperature, and some physical/mechanical statistics during deformation, with the following formalism. (1)

or

where is an array of measured properties including varying plastic strain εp, strain rate , forming temperature T, and M other experimentally observed physical properties varying progressively with deformation. An ideal phenomenological model generally consists of strain hardening function, strain rate sensitivity function, and thermal softening function. An exact definition of strain-hardening term is of significant importance to determine a basic shape of constitutive curve taken to be reference for the ensuing identification of strain rate and temperature dependences [26]. A non-exhaustive list of classic strain hardening functions including isolated and coupled ones with strain-rate and temperature as from published literature on phenomenological constitutive model is presented in Table 1. At high temperature or strain-rate levels, coupled effects of temperature and plastic strain, or strain-rate and plastic strain may show strong effects on the material strain-hardening behavior . Apart from strain hardening formalism covered in Table 1, any weighted combination of the strain hardening terms listed in Table 1 can also be adopted to formulate it in multiplicative or additive form.

(5)

(6)

A dependence-based integrated methodology for constitutive identification is proposed with the following specific procedures. Step 1: The isolated strain hardening function form is chosen from those listed in Table 1, or any weighted combination of those strain hardening functions such as Eq. (2) according to the shape of experimental data curve.

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Step 2:

is verified for feasibility by (a) the quality of fit performed for quasi-static test data at reference temperature and strain rate and (b) the relative error of the fit initial yield point σ(εp =0) value against the offset yield point within a range of 10%. If a poor fit quality (Section 3.3.3) or the relative error of more than 10% is arrived at, go to step 1 for modifying the function form until the two criteria above are both fulfilled.

(2)

Step 3: The thermal softening function form h(T) is chosen among those listed in Table 3 or from any weighted combination of those thermal softening functions.

The parameter A in Table 1 is typically considered initial yield point at quasi-static loading condition taken as the reference for temperature and strain-rate, whereas the yield point of most engineering materials cannot be precisely defined through stress-strain curve shape. The assumption that an offset yield point at plastic strain of 0.1 or 0.2% is chosen arbitrarily as the candidate for the parameter A might be questionable [51]. It is recommended to consider A as a free variable to be fit, and then the candidate for the parameter A is verified for feasibility by judging the relative error of it against the offset yield point within a range of 10%. The most common forms of empirical strain-rate sensitivity functions are summarized in Table 2. The strain rate sensitivity is introduced into strain hardening laws by the addition or product of flow function by strain hardening law in quasi-static mode taken as reference.

Step 4: The feasibility of h(T) is verified by performing multi-objective fit (Section 3.2) of the trial constitutive model in Eqs. (7) or (8) for multiple quasi-static curves of varying flow stress with forming temperatures under different plastic strains. If a poor fit is arrived at, go to Step 3 and revise it. (7)

or

(3)

or

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For thermal-softening function, apart from the simple and well-used Johnson-Cook model, other trial thermal-softening ones h(T) can be chosen from those reported in the literature, as listed in Table 3, or modified accordingly on that basis. Significant correlation between strain-rate and temperature is shown in mechanical behavior, particularly at high strain-rate levels. Hence, thermal softening function is often involved in formulating a constitutive model in the multiplicative form with strain-rate sensitivity function for general-purpose applications as follows.

2. Dependence-based integrated methodology for constitutive identification

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Step 5: The coupled strain hardening function form with temperature is chosen from Table 1, or any weighted combination of those coupled ones with temperature.

(4)

Step 6:

The additive strain rate flow function in Eq. (3) is assumed to be a new strain-hardening function, while the multiplicative one in Eq. (4) corresponds to the evolution of yield surface with strain rate. The coupled effect of strain rate and temperature is often introduced into the strain-rate sensitivity function .

is determined by seeing if the trial constitutive model in Eqs. (9) or (10) can be fit equally well to multiple quasi-static test curves of flow stress against forming temperatures using the weighted multi-objective identification strategy. Otherwise, go to Step 5 and revise it.

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Table 1 Typical strain hardening functions in common phenomenological models. k(εp) = A + kun

ki

A

1

A

2

σps

14

0

No

k(εp) = A∙kun

15

A

21 22 23 24 25 26 27 28 29 30 31 32 33 34

Shin model [29]

Polynomial model Jeong model [30] Khan–Huang–Liang model [31] Voce model [32] Samanta model [33]

See above

σp0

A

− (σps − σpo)exp( − nεp/ε0)

Microstructural evolution model [22] Models considering nanocrystalline evolution [34] Model considering size effect [35] Voce-Kocks model [36] Sellars model [37]

n

(σpm − σp0)[1 − exp( − nεp)] n K1εp

Ludwigson [38]

+ exp(K2 + n1εp)

kun

Comment

A∙kun isolated form

Generalized Swift model [39]

n

(1 + ε/b)

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18 19 20 No ki

Generalized Ludwik model [28]

n

Blnεp

9 10 11 12

17

Ludwik model [27]

n

Bεp

− Bexp ( − nεp)

8

16

A + kun isolated form

B1[1 − exp( − B2εp)]

4 5 6 7

ki

kun

B(εp + ε0)

3

13

Comment

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No

Hollomon model [40], gives poor fit in low strain

n εp

Avrami-Type Model [41]

n

[1 − exp( − Bεp)] tanh(Eε/A) See above See above

A

See above

Prager model for idealized plasticity [42] Models considering grain evolution Models considering nanocrystalline grain evolution Comment A + kc or A∙kc coupled form with rate and temp

kc

variation of strain hardening with grain size [43] Coupled strain, temperature and rate model [34] KLF model [44], consider grain evolution Thermal-softening effect on strain hardening 1 [18] Thermal-softening effect on strain hardening 2 [45]

− B1exp[B2 + B3(T − Tr)εp]

Thermal-softening effect on strain hardening 3 [46] Thermal-softening effect on strain hardening 4 [47]

*p

kun[1 − T ] kunexp(nεp)

Misiolek model [48]

kunexp ( − nεp)

Coupled strain hardening-softening model [49] Coupled rate-temperature-strain model 1 [50] Coupled rate-temperature-strain model 2 [8]

kunh(T)

Coupling with temperature in Table 3 Coupling with strain rate in Table 2

Coupling with rate and temp in Tables 2 and 3

The isolated strain-rate sensitivity function is chosen from Table 2 or any weighted combination of those isolated strain-rate sensitivity functions.

(9)

or

Step 8:

is determined by fitting the trial constitutive model in Eqs. (11) or (12) formulated by the chosen function to multiple dynamic curves of flow stress with strain-rate at the reference forming temperatures, at several different plastic strains of (no significant temperature

(10)

Step 7:

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Ceramics International xxx (2018) xxx-xxx Table 3 Typical thermal softening functions in common phenomenological models. No.

No.

Comment

gi

B

isolated form with temperature

1

1

1

2

1

3

0

5

gi

B

6

1

10

0

11 12

See above

where or

6

[T/Tr]

−m

m

[(Tm − T)/(Tm − Tr)]

7 8 9-

or

or

or

.

, or

or

*

T

1 + λ(e

Ta/Tm

−e

e a m) 1 − D(T − Tr)

)/(e −

exp ( − D[(T − Tr)/(Tm − m

Johnson-Cook thermal softening model [6] Power law model [61]

Tr)] )

Khan model [50] Thermal softening model [62], suitable for HCP materials Linear model [63]

Thermal softening model exponential model 1 [64] Thermal softening model exponential model 2 [65] Thermal softening model exponential model 3 [66] Thermal softening model exponential model 4 [67]

The coupled strain-rate sensitivity function with temperature is chosen from Table 2 or any weighted combination of those strain-rate sensitivity functions. Step 10:

is verified for feasibility by performing a multi-objective fit of the trial constitutive model in Eqs. (13) or (14) to multiple dynamic curves of varying flow stress at several high plastic strain values with strain-rate at different forming temperatures. If a good fit is achieved, the basic form of constitutive model is formulated by the above identification procedures. Otherwise, go to Step 9 and revise it. (13)

or

;

(14)

Step 11: A trial full constitutive model , formulated using dependence-based integrated methodology, is fit to all the experimental data point using a weighted multi-objective identification strategy such that a set of material parameter values are identified. The model feasibility is validated by measuring the overall fit quality, average absolute relative error and asymptotic fit standard error. If a good overall quality of fit is available, the identification processes of constitutive model and its material parameters are finished. Otherwise, go to Steps 3, 5, 7 or 9 for the corresponding modification of strain hardening, strain-rate sensitivity or thermal softening functions, according to the relative fit standard parameter error RFSPE in Eq. (15), equal to the fit standard parameter relative errors FSPE divided by the parameter vector P, until the specified error criteria above are fulfilled simultaneously.

(11)

(12)

Step 9:

m

1 − [(T − Tr)/(Tm − Tr)]

T /T

5

rise), using multi-objective identification strategy. Otherwise, go to Step 7 and revise it.

or

Comment

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9

4

coupled form with temperature Variable strain rate sensitivity [56] Reduced rate strengthening at high rate [57] Enhanced rate strengthening model 1 [58] Coupled rate-temp ZA model for BCC materials [8] Coupled rate-temp exponential model [59] Coupled rate-temp power model [60] Coupling with temperature

7 8

3

Johnson-Cook rate dependent model [6] Cowper-Symonds model [52] Power law model [53] Wagoner model [54] Rate sensitivity exponential model [55] Comment

4

No.

2

h(T)

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Table 2 Typical strain-rate sensitivity functions in common phenomenological models.

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but varying with plastic strain history, as vividly illustrated in Fig. 2. Thus, the artificial strengthening of thermal softening effect will lead to an inaccurate estimate of strain-rate sensitivity coefficient. Realistically, varying forming temperatures and strain rates with deformation (see the Appendix A) limits the identification of constitutive parameters using classical non-linear regression strategy. Since the nonconstant nominal/average strain rates and forming temperatures during process are considered in accurate identification of constitutive parameter, an improved and generalized parameter identification based on the weighted multi-objective methodology is proposed to find a more accurate set of material parameters over the entire span of quasi-static and dynamic modes.

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where J and W are defined only after in the manuscript. 3. Weighted multi-objective strategy for material parameters identification 3.1. Identification of material parameters: background The commonly used methods for material parameter determination are divided into two categories: inverse analysis and non-linear regression. Inverse analysis is a minimization method of determining the material parameters by solving the least-square sum of the deviation between experimental data in a global range and the model prediction values at the same strain or by performing finite element analysis for comparable simulation results to experimental observation [68,69]. This non-linear regression is to obtain multiple sets of parameter values under localized experimental values in deformation history, through univariate analysis of constitutive model and then to average them to find a final set of parameters [26]. The nonlinear regression approach for parameter identification is implemented successively through the following routines:

3.2. Weighted multi-objective identification methodology

Together with the above dependence-based integrated methodology for constitutive identification, an improved multi-objective identification methodology for material parameters is proposed to arrive at a more accurate, reliable, and generalized set of material parameters. To do this, this paper is based on the criteria from Ref. [71] and suggests a modified version of some criteria for the formulation of objective function as follows.

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Criterion 1 The measurement errors of experimental data in one deformation mode should be taken into account in the formulation of objective function.

– The material parameters for work-hardening function can be evaluated through nonlinear regression from quasi-static test, i.e. at reference strain-rate, forming temperature, and other physical or mechanical properties. – The parameters for strain rate sensitivity can be identified by performing regression analysis upon mathematical transformation of dynamic test data in the reference temperature under different plastic strain, and averaging the inversely-transformed parameter values under different plastic strain to obtain final evaluation of strain rate sensitivity parameters. – The parameters for thermal softening can be estimated using the similar approach to the one for determining strain rate sensitivity parameters. – The parameters for characterizing the dependence of flow stress on other physical or mechanical processes, can be calculated by performing regression analysis of flow stress against this physical or mechanical variable under different levels of one independent variable such as strain, strain-rate, or temperature, and later performing nonlinear fit of the physical or mechanical variable against the independent variable. Its identification process is actually more complex due to the coupled effects of it and other independent variables.

Criterion 2 For single curve, all the experimental data points should be accounted for and have equal pressure to be optimized during parameters determination. When optimizing multiple curves corresponding to different deformation modes, it should be ensured that all experimental curves are endowed with equal optimization opportunities, with independence of the number of experimental data in any deformation mode. If involving multi-sub-objectives, individual objective function should be capable of dealing with incorporation of multiple sub-objectives by assigning equal opportunity to be optimized for individual sub-objective, independently of the number of data points in each sub-objective. Criterion 3 Continuity condition of multi-objective function should be fulfilled by means of progressive evaluation of fitting quality

The constant (average) strain-rate experimental data are employed for determining material parameters for strain-hardening and thermal softening functions through the conventional nonlinear regression approach. The isothermal uniaxial mechanical test is a commonly used method for determining the flow stress of a material. The quasi-static test is usually carried out on a universal testing machine. However, the motion is applied at constant travel speed (equal to the product of nominal strain rate and gauge length hw) rather than constant nominal strain-rate on the universal testing machine. In the case of large deformation, the deviation from nominal strain-rate have significant influence upon determination of strain-hardening model of the materials showing the sensitivity of flow stress on strain rate even in quasi-static mode. Moreover, the temperature rise during deformation is not constant

1 Fig. 2. Computed temperature rise of SiCp /Al composites from plastic strain at 6000 s− and 100 °C.

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since the gradient-based optimization algorithm cannot be executed in a discontinuous function.

give equal opportunity to each objective function corresponding to one loading mode, the normalization is performed on all objective functions by considering measurement error. Generally, the measure deviation of all experimental points in one deformation mode should be of either constant measurement error, or proportional error to flow stress . Under the assumption of constant measurement error in each loading mode, since the measurement error are unknown, the covariance analysis can be performed to arrive at the measurement error [72]. The measurement error in any deformation

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Criterion 4 The weighting factors, used for allocating equal opportunity to be optimized for each curve, should be assigned automatically based on available statistical evaluations. A general methodology for inverse identification of material parameters P is to minimize the difference between experimentally observed material behavior and constitutive model predictions, with the following basic formalism.

mode is evaluated from the following fitting for the ith experimental data point in the jth deformation mode.

(16) s.t. (17)

where k is scaling factor. The weighting factors obtained by variance analysis of measurement errors are here introduced to formulate multi-objective function.

(18) where P is a vector of parameters to be fitted, Nj the number of exper-

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imental points under the jth loading condition, and M the number of loading conditions. The superscripts e and m label the experimental and constitutive-based calculated values, respectively. The accuracy and reliability of inverse analysis strategy for parameter identification are dependent upon the information involved in the objective function, which is arrived at by incorporation of weighting factors in the formulation of multi-objective function [71]. A proper definition of multi-objective function, thus, will result in more accurate identification for constitutive parameters. As can be seen in Fig. 3, the measurement deviation/error of flow stress obtained from dynamic experiments form mean curve at different forming temperatures might be of different scales, i.e. 21 MPa at low temperature and 8 MPa at high temperature within 99% confidence interval. It should be noted that if the measurement errors in different loading modes were of different orders of magnitude, a set of estimates for material parameters might minimize the specific objective function in one deformation mode with larger measurement error and not all objective functions. This would lead the fit material model to perform well in one loading mode but not in others. To

The assignment of weighting factors based on measurement error covariance to different deformation modes also contributes to satisfy the Criteria 1 and 4. Simultaneously, the influence of those individual objective functions with poor quality of fit on multi-objective minimization process is reduced by the assignment of this weighting factor, denoting individual fit standard error. More precisely, during multi-objective optimization process for parameter identification, it is also important to note that equal opportunities to be optimized is given to each objective function corresponding to one mechanical test. Otherwise, the fit constitutive model may behave well in predicting mechanical responses in one loading condition but not in others. This is very likely if the number of experimental points and/or number of active constitutive parameters (less parameters in quasi-static mode) in different loading conditions are of magnitude. The weighting factors in the jth deformation mode considering the number of experimental points and number of active constitutive parameters should be formulated in terms of Criteria 2 and 4, as follows. (21)

Therefore, coupled with the assignment of weighting factors in Eqs. (20) and (21), the multi-objective function in Eq. (16) can be reformulated by (22)

The above multi-objective function can be rearranged to be non-dimensionalized, taking the following relative formalism

Fig. 3. Schematic of flow stress measurement deviation at different forming temperatures.

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of

W is weighting matrix, of which the weighting component for the ith experimental data point in the jth deformation mode with respect to the kth material parameter can be formulated in the multiplicative form of weighting factors in Eqs. (20), (21) and (27) as

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(23) Form the Eq. (23), it implies that this relative multi-objective function normalized for parameter identification can be used for concurrent optimization of other available material behaviors with different units. The sub-objective functions in quasi-static and dynamic modes can be derived from Eq. (22) as

J is Jacobian matrix, of which the component Jik is the partial deriv-

ative of the ith experimental point parameter Pk in P, as expressed by

(24)

with respect to the kth material

T

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may lead to the unachievable inversion of [J WJ + λI]. An alternative approach aiming at defining a suitable λ for Levenberg's algorithm was suggested by Nielsen [73]. It is proved that the Levenberg-Nielsen algorithm is of better performance on convergence and time cost, compared to Levenberg–Marquardt algorithm. An effective initial value of the damping factor λ0 is recommended as follows.

To be capable of representing the quasi-static and dynamic material behavior simultaneously and equally well, an equally weighted bio-objective function is formulated in an additive form of quasi-static and dynamic objective functions in Eqs. (24) and (25) as (26)

(32)

where τ is an initial guess for which the recommended value range is 6 to 10− 3, beyond which the poor guess is made [74]. from 10− If the metric , the damping factors are iteratively updated according to the following criteria,

(27)

and

(31)

3.3.2. Initialization and update of damping factor λ and step length h If the damping factor λ in Eq. (29) is large, Levenberg's algorithm

(25)

with

(30)

(33)

and if Q(hi) ≤ ϵ4, then

(28)

(34)

The weighted multi-objective optimization for inverse identification for constitutive parameters is carried out by the Levenberg algorithm combined with the suggested damping factor by Nielsen [73], with good performance on convergence and time cost.

where νi is scaling factor, Q(hi) is a ratio between the actual and expected improvement in the objective function [75].

3.3. Numerical implementation

(35)

ϵ4 is a specified threshold controlling the acceptance of step update

3.3.1. Algorithm The weighted multi-objective optimization for inverse identification for constitutive parameters is carried out by the Levenberg algorithm for solving the nonlinear least-square problem in Eq. (26), as follows.

h. If

, indicating Pi + hi is superior to Pi, Pi is replaced by

Pi + hi. Otherwise, the damping factor is updated according to Eq. (34), and the algorithm proceeds to the next iterative process.

(29)

3.3.3. Error analysis The coefficient of determination, R2 , is used for evaluating the overall quality of fit, which provides a direct measure of how well experimental observations are replicated by fitting model.

Here Pi + 1 = Pi + hi, where Pi and hi are the fit vector and step length vector of the parameters at the iteration step s,

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(36)

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with (37) It is worth noting, additionally, that even a higher R2 may not necessarily imply better predictability and reliability since fitting model tends to be biased toward lower or higher evaluation [76]. Thus, the average absolute relative error AARE and asymptotic fit standard error AFSE are used to give an unbiased statistical error measure of model predictability and reliability, with the following formalism.

Fig. 4. Microstructure of Al6061/SiCp/15 composites.

height to perform temperature measure. The specimen was preheated at 4 K/s up to a desired temperature, and then held for 3 min at this forming temperature. The cylindrical samples of Φ6 mm × 9 mm in size were 1, at nominal forming temcompressed at constant strain rate of 0.001 s− peratures of 25, 100, 200, 300 and 400 °C in a vacuum chamber. The dynamic compressive tests were performed for the cylindrical specimen of Φ4 mm × 4 mm on a SHPB apparatus, at the nominal forming temperatures of 25,100, and 200 °C, at the nominal strain rates of 1. The SHPB experimental details are pre1000, 2000, 5000 and 7000 s− sented in Ref. [78].

(38)

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(39) The fit standard parameter errors FSPE, as a measure of experimental data variability effect on the parameters variability, can be calculated by taking the square root of the leading diagonal elements of the variance-covariance matrix of the parameter vector P.

4.3. Results

(40)

Fig. 5(a) and (b) respectively illustrate the quasi-static and dynamic true stress-strain curves of Al6061/SiCp /15 composites at different forming temperatures, with the offset yield point Rp 0.2 = 264.0 MPa. From quasi-static curves it is found that with varying forming temperature, the strain hardening behaviors show temperature dependence, and at high temperature than 400 °C, almost no strain hardening arise. The significant thermal softening effect of flow stress in quasi-static mode is temperature dependent, and so it is in dynamic loading mode. The strain-rate sensitivity of flow stress at identical forming temperature in dynamic mode are observed, especially at low forming temperature. Besides, the greater fluctuation of flow stress in dynamic mode than that in quasi-static mode may be ascribed to the nature of variable strain-rates in dynamic loading process.

From the methodology view of point, only by a fewer number of tests where both strain-rate and temperature can vary simultaneously, can the determination of material parameters be achieved using multi-objective identification strategy, compared to classical identification strategies. 4. Case study 4.1. Materials

SiC particulate reinforced Al matrix composites have received wide attention in aerospace, military, automobile, and civil manufacturing fields, due to high specific strength and stiffness, wear resistance, fatigue resistance, and tailored properties above, over conventional aluminum alloys [77]. The SiCp /Al composites employed for constitutive modelling is15% volume fraction SiC particulate reinforced Al 6061 matrix composites (Al6061/SiCp /15 composites) processed using pressure infiltration method. Using 3D Laser Scanning Microscope VK-X200, the micrograph of Al6061/SiCp/15 composites is illustrated in Fig. 4, showing the microstructure of SiC particulate with some clustering and no preferred orientation in Al matrix alloy. The reinforcement phase SiC particulates with many sharp corners are approximatively polyhedral and the average particulate size are about 5 µm.

4.4. Constitutive modelling Combined with the weighted multi-objective strategy for material parameters identification, the proposed dependence-based integrated methodology is employed for identifying constitutive model form and its material parameters for SiCp /Al composites. Single-objective fit of the suitable strain hardening functions chosen in Table 1 for Al6061/ SiCp /15 composites is performed for quasi-static mechanical data at 3 s− 1 and 25 °C. Meanwhile, an efficient set of loading conditions of 10− initial estimates are arrived at from a single-objective fit of the quasi-static data. Table 4 presents single-objective fit results of the suitable strain hardening functions. From a fit-quality view of point, the Shin, Polynomial, Jeong, Voce models can provide the capability of predicting strain hardening behavior in quasi-static mode well. Fig. 6 shows the experiment data and models prediction of flow stress-plastic strain of Al6061/ 3 − SiCp /15 composites at 10− s 1 and 25 °C. The prediction range of the

4.2. Experimental

The quasi-static compression experiments were conducted on Gleeble 3500 thermomechanical testing machine. The S-thermocouple was joined to the location near half the sample

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gives

with

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(41)

(42)

where A is initial yield stress, B1 and B2 are strain-straining coefficients,

Wd is weighting coefficient, G is thermal softening coefficient, Tref and Tmelt are reference temperature and melting temperature, respectively.

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It yields a feasible fit results with the average absolute relative error of 6.6%, asymptotic fit standard error of 20.12 MPa, and R2 of 98.50%. The comparison of experimental data and identified model prediction is shown in Fig. 7(a). The overestimated flow stress by the identified model implies the temperature dependency of strain hardening behavior. Therefore, thermal softening effect is to be introduced to the mechanical characterization of strain hardening behavior. The chosen coupled strain hardening function form with temperature is not determined using weighted multi-objective identification method until it can be fit equally well to all quasi-static experimental curves, with an acceptable degree of fit. The multi-objective inverse identification yields a weighted thermal softening function form in Eq. (43), with (R2 , AARE, AFSE) being (99.70%, 4.30%, 7.7866 MPa). Comparison of experimental data and model prediction curves in quasi-static loading modes is presented in Fig. 7(b), illustrating the flexibility and accuracy of this weighted multi-objective identification strategy.

Fig. 5. True stress-plastic strain curves of Al6061/SiCp /15 composites at different forming temperatures (a) under quasi-static loading, and (b) under quasi-static loading.

Polynomial model is only within a narrow experimental plastic strain, and not extended to a large strain beyond the experimental data. The highest degree of fit for Jeong model is ascribed to more parameters to be fit, increasing the constitutive complexity. Voce and Shin models differ in equation form but coincide in nature. Hence, Voce or Shin models can be chosen for the function formalism describing strain hardening behavior of Al6061/SiCp /15 composites. The experimental data of flow stress-forming temperature at different plastic strains in quasi-static mode are chosen as many as possible for multi-objective identification of thermal softening function. The optimum form of thermal softening function

(43)

There exists a small deviation between experimental data and model prediction at 100 °C, and the experimental flow stress curve at 100 °C shows stronger strain hardening than

Table 4 Single-objective fit results of the suitable strain hardening functions. Model

Ludwik G-Ludwik Shin

Polynomial Jeong Voce

Fit equation

Overall quality of fit

0.1952

231.7 + 229.4εp

0.0000457 + 448.3(εp + 0.0007156)

0.07298

290.9 + 105.7[1 − exp( − 22.8εp)] −

4 1.7E5εp

+

3 1.04E4εp

400.8 − 125.9exp( −

−

2 2.33E4εp

0.7306 10.78εp )

R2

AARE

AFSE

σ(εp = 0)

95.14%

2.072%

5.661

231.7

97.51%

1.350%

4.055

264.2

99.30%

0.631%

2.146

290.9

99.92%

0.177%

0.707

274.9

99.30%

0.631%

2.146

290.9

99.51%

+ 2397εp + 290

396.5 − 105.6exp( − 22.8εp)

10

0.487%

1.797

290.0

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objective inverse identification strategy, with the fit results illustrated in Fig. 7(d). A best fit to the above experimental data with a high fit quality of R2 = 98.94%, AARE= 1.38%, and AFSE = 4.74 MPa, is identified as (45)

Thus, a trial constitutive model for predicting material behaviors of Al6061/SiCp /15 composites in static and dynamic modes is established in Eq. (46). Since the above identification process of material constitutive formalism is aimed at the fit to local experimental data but not all ones, the final objective is to find a set of material parameters arrived at using multi-objective inverse identification strategy by fitting the identified constitutive model well to all the experimental data.

(46)

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The quantitative identification results of fit upon R2 , average absolute relative error and asymptotic fit standard error are found to be 99.62%, 2.34%, and 8.9138 MPa, respectively. The material parameter of the identified constitutive model above for Al6061/SiCp /15 composites are presented in Table 5. The experimental flow stress-plastic strain curves in both quasi-static and dynamic loading modes are illustrated in Fig. 8, together with the corresponding simulations using the identified constitutive model. A good fit indicates that the proposed dependence-based integrated constitutive modelling methodology, together with multi-objective parameter identification strategy may provide a new methodology in characterizing material behaviors in quasi-static and dynamic modes equally well.

3 s− 1 and Fig. 6. Experiment and models prediction of flow stress-plastic strain at 10− 25 °C.

those at other temperatures. This special phenomenon may be due to the increase in the diffusion rate in this temperature range from 100 °C to 400 °C that make the diffusion become active. In order to reduce the temperature dependencies in mechanical characterization of strain-rate sensitivity behavior, the variation of flow stress at very small plastic strain of 0%, 0.2%, 0.5% and 1% with strain rate under reference temperature 25 °C are employed for identifying isolated strain-rate sensitivity function through multi-objective fit strategy. Fig. 7(c) illustrates the experimental data and multi-objective fit curves in dynamic loading modes. When the Cowper-Symonds rate-dependent function in Table 2 is applied in Eq. (44), it yields a better fit quality of R2 = 94.0%, AARE= 2.52%, and AFSE = 7.74 MPa, over other strain-rate sensitivity ones. Fig. 7(c) shows the experimental and multi-objective fit curves at small plastic strains in dynamic loading modes.

5. Conclusions

The phenomenological model is widely used for the characterization of plastic behavior of metals and alloys during forming and manufacturing. But the applications of phenomenological constitutive model in industrial simulation are often faced with some major difficulties such as the requirement of a larger number of mechanical tests over a wide loading range by the classical parameter identification strategy, determination of coupled relations among explanatory variables, and the ensuing hard identification job of material parameters involved in constitutive characterization. This paper suggests an improved methodology for dependence-based integrated development of constitutive model, together with multi-objective identification scheme of material parameters with the equal capability of predicting material behaviors in quasi-static and dynamic modes well. The automatic assignment of three weighting factors considering (a) different measure errors or individual fit standard errors in individual objective function corresponding to each test, (b) the number of experimental points and number of active material parameters under different loading conditions, and (c) assignment of equal optimization opportunity for quasi-static and dynamic sub-objective functions, is introduced in the formulation of multi-objective identification model to arrive at a more accurate, reliable, and generalized set of material parameters over the entire span of

(44)

As can be seen from the overestimated flow stress in Fig. 7(c) and a not-too-high R2 value, the effect of temperature dependency is to be incorporated into strain-rate sensitivity function to arrive at a high degree of fit. Hence the ensuing identification task is to find the suitable strain-rate sensitivity function with temperature dependencies for capturing general tendency of strain rate-dependence. Multiple dynamic curves of varying flow stress at higher plastic strain values of 10%, 15%, and 20% with strain-rate at forming temperatures of 25 °C, 100 °C and 200 °C are employed to identify specific coupled relation of strain rate hardening with temperature using multi-

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Fig. 7. Experimental and multi-objective fit curves (a) at different plastic strains within a large range, (b) in quasi-static loading modes, (c) at small plastic strains at 25 °C, and (d) at larger plastic strains under different temperatures. Table 5 Multi-objective fit results of the identified full model for Al6061/SiCp /15 composites. Para. Value FSPE

A

B1

B2

n1

Wd

n3

n2

D

m

G

314.2 1.306

74.45 1.245

24.84 0.787

1.387 0.105

0.8584 0.003

4.652 0.044

2.385 0.081

1,034,000 191,850

4.389 0.164

72.82 3.224

quasi-static and dynamic loading conditions, independently of a user. The performance of this dependence-based integrated constitutive modelling methodology and multi-objective parameter identification strategy is verified by illustrating the multi-objective fit to the experimental data of Al6061/SiCp /15 composites from quasi-static Gleeble and dynamic SHPB tests with the instantaneous strain rates variation and corrected plastic strain-related forming temperatures changes, where the classical parameter identification strategy cannot work. A good fit shows the feasibility of the proposed dependence-based integrated constitutive methodology and the qualification of weighted multi-objective identification strategy in characterizing overall material behaviors in both quasi-static and dynamic modes.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51575051) and National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2012ZX04003051-3). The authors are also grateful for the technical support from the School of Aeronautics at Northwestern Polytechnical University. Conflict of interest The authors declare no competing financial interests.

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Though the influence of the variation of strain rate on most engineering materials in quasi-static test is not considerably significant, it is of significant importance to incorporate strain rate effect into a correct determination of constitutive parameters in quasi-static mode. Hence under kinematic conditions of large displacements and large strains, the true strain rate for the ith experimental data point can be corrected from Eq. (A1). (A1)

where di is the displacement for the ith experimental data point. But

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if so, the classical identification strategy cannot be used in parameter identification of strain hardening function. In the classical identification strategy, only a constant (average) nominal strain rate, instead of practical strain rate corresponding to each data point in dynamic test, is employed during parameter identification process. To improve the fit precision of constitutive model, instantaneous strain rates in dynamic tests are chosen for optimization. In this case, the conventional nonlinear regression approach is limited due to nonconstant strain rate. An alternative quasi-static test without any correction is Gleeble thermomechanical testing, where the instantaneous strain rate can be kept nearly constant. It is worth noting that in isothermal uniaxial test when strain rate 3 s− 1, the heat generated by plastic deformation are dissipated < 10− timely due to sufficient deformation time, so the test at strain rate of 3 s− 1 can be taken as isothermal deformation. When strain less than 10− 3 s− 1, thermal-softening effect during deformation is rate exceeds 10− so considerable that non-isothermal deformation with temperature correction should be taken into account [70]. As the strain rate increases 1, the plastic work during deformation results in conversion of a to 1 s− considerable fraction of plastic strain energy into adiabatic heat in localized regime. The quantitative relation between plastic work ∆W and converted heat including dissipated heat can be given based on the first law of thermodynamics, as follows. (A2)

Here ∆W = ∫σdεp and Qw = ρCp∆T, where ρ and Cp are respectively

density and specific heat, Δεp plastic strain during deformation. η is the

Taylor–Quinney coefficient that stands for the fraction of conversion of mechanical work into heat, typically assumed to be 0.9–0.95, whereas the remaining fraction used for microstructure change (grain evolution, dynamic recrystallization, phase transformation, etc.). The adiabatic correction factor α, represents the fraction of adiabatic heat remaining in test specimen due to heat dissipation into die and environment. In terms of strain rate range, the adiabatic correction factor varying with strain rate is defined as [70]

Fig. 8. Comparison of experimental and simulations using multi-objective identification constitutive model in the modes of (a) quasi-static loading, (b) dynamic loading at 25 °C, and (c) dynamic loading at elevated temperatures.

(A3)

with

Author contributions

J.F.X. and L.J.X. designed this research. J.F.X.and F.N.G. carried out quasi-static and dynamic mechanical tests. J.F.X. wrote the Matlab codes of multi-objective identification strategy for material parameters. J.F.X., L.J.X., J.Y., S.Q.P. and X.B. Wang wrote this manuscript.

(A4)

where hw is the workpiece height, dd the distance from die surface to die interior where the temperature keeps constant, κinterface heat transfer

13

coefficient

of

work/die

interface,

κw

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and κd thermal conductivity of workpiece and die, respectively. From

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Methodology for dependence-based integrated constitutive modelling: An illustrative application to SiCp /Al composites Junfeng Xianga , Lijing Xiea , b , ⁎ , Feinong Gaoa , Jie Yia , Siqin Panga , b , Xibin Wanga , b a b

School of Mechanical Engineering, Beijing Institute of Technology, 100081 Beijing, China Key Laboratory of Advanced Machining, Beijing Institute of Technology, 100081 Beijing, China

ABSTRACT

Keywords: Constitutive modelling Multi-objective Parameter identification SiCp /Al composites

In industrial forming and machining process, the large plastic deformation of material takes place in wide loading ranges of strain-rate and forming temperature. A satisfactory modelling of quasi-static and dynamic material behaviors is of great importance for understanding physical process and processes optimization. A dependence-based integrated methodology, together with an improved weighted multi-objective parameter identification strategy is presented for the development of phenomenological constitutive model and the parameter identification using experimental data from quasi-static and dynamic tests with instantaneous strain rate variations and plastic strain-related temperature changes. The improved multi-objective parameter identification model is reformulated by introducing three weighting factors for valuing different measure errors and fit standard errors in individual objective function corresponding to each test, considering the sampling point number and active material parameter number under different loading conditions, and balancing optimization opportunity of quasi-static and dynamic sub-objective functions. The methodology is verified for feasibility through illustrative constitutive identification for SiCp /Al composites. This may provide a methodology of constitutive modelling for predicting material behaviors in quasi-static and dynamic modes equally well.

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ARTICLE INFO

1. Introduction

In manufacturing industry, almost all material forming and machining processes consist of deforming plastically workpiece under the forces applied by a die or tool [1]. Understanding of material flow behavior under different deformation modes is gaining critical importance in planning of material forming and manufacturing processes, selection of die/tool, design feasibility of final products [2–4]. A schematic of the dependence of analytical modelling and finite element modelling on constitutive model in material forming and machining processes is illustrated in Fig. 1. Therefore, it is necessary to develop/establish constitutive model reflecting the yield criteria, hardening laws and flow rule of the material under com

⁎

bined loading conditions of deformation rate and forming temperature. The accuracy of numerical simulation available for industrial applications depends not only upon the applicability and flexibility of material constitutive model, but also the scheme applied to determine material parameters [5]. With the continuous emergence of new materials, classical constitutive models (e.g., Johnson-Cook [6], Arrhenius-type [7], Zerilli-Armstrong [8], etc.), much less so-called unified constitutive models that tend to describe a given crystalline structure or crystalline system, cannot accurately predict the flow behaviors of those new materials in a wide loading range. These pose encountering challenges to the determination of material constitutive models. For improving the applicability and flexibility of the above classical constitutive models, it is necessary to develop or mod

Corresponding author at: School of Mechanical Engineering, Beijing Institute of Technology, 100081 Beijing, China. Email address: [email protected] (L. Xie)

https://doi.org/10.1016/j.ceramint.2018.03.257 Received 21 December 2017; Received in revised form 15 February 2018; Accepted 28 March 2018 Available online xxx 0272-8842/ © 2018.

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Fig. 1. Incorporation of constitutive model in analytical modelling and finite element modelling of material forming and machining processes.

ify advanced constitutive models based on the classical phenomenological constitutive models by introducing the coupled effects (e.g., between strain-rate and temperatures [9], between strain and temperature [10], between other physical or mechanical properties and strain [11,12], and among strain, strain-rate and temperature [13]), some special mechanical behaviors (e.g., strain-softening at high strain [14], scale effect [15], ratcheting effect [16], anisotropy [17], plastic strain gradient [18], etc.), and other experimentally observed physical or mechanical properties varying with deformation such as density in foam metals [19], microscopic damage in composites [20], phase transformation [21], and grain evolution [8,22], etc. This would also lead to the complexity of constitutive model involving more material parameters, and increase the experimental cost for developing constitutive model and identifying its material parameters, of which some can be attained through microscopic observation [20], and some can be also fit

to numerous experimental data in quasi-static and dynamic modes [9–12]. Therefore, the major difficulties for widespread application of phenomenological constitutive model for industrial simulation consist in a larger number of mechanical tests over a wide range of loading conditions needed for using classical identification methodology of constitutive model, identification of coupled relations among explanatory variables, and the ensuing hard identification job of material parameters involved in constitutive characterization [23,24]. The aim of this paper is to present an improved methodology for dependence-based integrated constitutive modelling, together with multi-objective material parameters identification strategy. Using an illustrative, yet complex example for SiC particulate reinforced aluminum matrix composites using experimental data from quasi-static and dynamic tests, the methodology is verified for feasibility by comparing the overall fit quality of the identified constitutive model to experimental data.

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A phenomenological constitutive model is actually an empirical relationship for mechanical characterization of complex material responses in different loading regimes [25]. The general formalism of a one-dimensional phenomenological model is to represent the dependence of flow stress on strain, strain-rate, temperature, and some physical/mechanical statistics during deformation, with the following formalism. (1)

or

where is an array of measured properties including varying plastic strain εp, strain rate , forming temperature T, and M other experimentally observed physical properties varying progressively with deformation. An ideal phenomenological model generally consists of strain hardening function, strain rate sensitivity function, and thermal softening function. An exact definition of strain-hardening term is of significant importance to determine a basic shape of constitutive curve taken to be reference for the ensuing identification of strain rate and temperature dependences [26]. A non-exhaustive list of classic strain hardening functions including isolated and coupled ones with strain-rate and temperature as from published literature on phenomenological constitutive model is presented in Table 1. At high temperature or strain-rate levels, coupled effects of temperature and plastic strain, or strain-rate and plastic strain may show strong effects on the material strain-hardening behavior . Apart from strain hardening formalism covered in Table 1, any weighted combination of the strain hardening terms listed in Table 1 can also be adopted to formulate it in multiplicative or additive form.

(5)

(6)

A dependence-based integrated methodology for constitutive identification is proposed with the following specific procedures. Step 1: The isolated strain hardening function form is chosen from those listed in Table 1, or any weighted combination of those strain hardening functions such as Eq. (2) according to the shape of experimental data curve.

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Step 2:

is verified for feasibility by (a) the quality of fit performed for quasi-static test data at reference temperature and strain rate and (b) the relative error of the fit initial yield point σ(εp =0) value against the offset yield point within a range of 10%. If a poor fit quality (Section 3.3.3) or the relative error of more than 10% is arrived at, go to step 1 for modifying the function form until the two criteria above are both fulfilled.

(2)

Step 3: The thermal softening function form h(T) is chosen among those listed in Table 3 or from any weighted combination of those thermal softening functions.

The parameter A in Table 1 is typically considered initial yield point at quasi-static loading condition taken as the reference for temperature and strain-rate, whereas the yield point of most engineering materials cannot be precisely defined through stress-strain curve shape. The assumption that an offset yield point at plastic strain of 0.1 or 0.2% is chosen arbitrarily as the candidate for the parameter A might be questionable [51]. It is recommended to consider A as a free variable to be fit, and then the candidate for the parameter A is verified for feasibility by judging the relative error of it against the offset yield point within a range of 10%. The most common forms of empirical strain-rate sensitivity functions are summarized in Table 2. The strain rate sensitivity is introduced into strain hardening laws by the addition or product of flow function by strain hardening law in quasi-static mode taken as reference.

Step 4: The feasibility of h(T) is verified by performing multi-objective fit (Section 3.2) of the trial constitutive model in Eqs. (7) or (8) for multiple quasi-static curves of varying flow stress with forming temperatures under different plastic strains. If a poor fit is arrived at, go to Step 3 and revise it. (7)

or

(3)

or

PR OO F

For thermal-softening function, apart from the simple and well-used Johnson-Cook model, other trial thermal-softening ones h(T) can be chosen from those reported in the literature, as listed in Table 3, or modified accordingly on that basis. Significant correlation between strain-rate and temperature is shown in mechanical behavior, particularly at high strain-rate levels. Hence, thermal softening function is often involved in formulating a constitutive model in the multiplicative form with strain-rate sensitivity function for general-purpose applications as follows.

2. Dependence-based integrated methodology for constitutive identification

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Step 5: The coupled strain hardening function form with temperature is chosen from Table 1, or any weighted combination of those coupled ones with temperature.

(4)

Step 6:

The additive strain rate flow function in Eq. (3) is assumed to be a new strain-hardening function, while the multiplicative one in Eq. (4) corresponds to the evolution of yield surface with strain rate. The coupled effect of strain rate and temperature is often introduced into the strain-rate sensitivity function .

is determined by seeing if the trial constitutive model in Eqs. (9) or (10) can be fit equally well to multiple quasi-static test curves of flow stress against forming temperatures using the weighted multi-objective identification strategy. Otherwise, go to Step 5 and revise it.

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Table 1 Typical strain hardening functions in common phenomenological models. k(εp) = A + kun

ki

A

1

A

2

σps

14

0

No

k(εp) = A∙kun

15

A

21 22 23 24 25 26 27 28 29 30 31 32 33 34

Shin model [29]

Polynomial model Jeong model [30] Khan–Huang–Liang model [31] Voce model [32] Samanta model [33]

See above

σp0

A

− (σps − σpo)exp( − nεp/ε0)

Microstructural evolution model [22] Models considering nanocrystalline evolution [34] Model considering size effect [35] Voce-Kocks model [36] Sellars model [37]

n

(σpm − σp0)[1 − exp( − nεp)] n K1εp

Ludwigson [38]

+ exp(K2 + n1εp)

kun

Comment

A∙kun isolated form

Generalized Swift model [39]

n

(1 + ε/b)

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18 19 20 No ki

Generalized Ludwik model [28]

n

Blnεp

9 10 11 12

17

Ludwik model [27]

n

Bεp

− Bexp ( − nεp)

8

16

A + kun isolated form

B1[1 − exp( − B2εp)]

4 5 6 7

ki

kun

B(εp + ε0)

3

13

Comment

PR OO F

No

Hollomon model [40], gives poor fit in low strain

n εp

Avrami-Type Model [41]

n

[1 − exp( − Bεp)] tanh(Eε/A) See above See above

A

See above

Prager model for idealized plasticity [42] Models considering grain evolution Models considering nanocrystalline grain evolution Comment A + kc or A∙kc coupled form with rate and temp

kc

variation of strain hardening with grain size [43] Coupled strain, temperature and rate model [34] KLF model [44], consider grain evolution Thermal-softening effect on strain hardening 1 [18] Thermal-softening effect on strain hardening 2 [45]

− B1exp[B2 + B3(T − Tr)εp]

Thermal-softening effect on strain hardening 3 [46] Thermal-softening effect on strain hardening 4 [47]

*p

kun[1 − T ] kunexp(nεp)

Misiolek model [48]

kunexp ( − nεp)

Coupled strain hardening-softening model [49] Coupled rate-temperature-strain model 1 [50] Coupled rate-temperature-strain model 2 [8]

kunh(T)

Coupling with temperature in Table 3 Coupling with strain rate in Table 2

Coupling with rate and temp in Tables 2 and 3

The isolated strain-rate sensitivity function is chosen from Table 2 or any weighted combination of those isolated strain-rate sensitivity functions.

(9)

or

Step 8:

is determined by fitting the trial constitutive model in Eqs. (11) or (12) formulated by the chosen function to multiple dynamic curves of flow stress with strain-rate at the reference forming temperatures, at several different plastic strains of (no significant temperature

(10)

Step 7:

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Ceramics International xxx (2018) xxx-xxx Table 3 Typical thermal softening functions in common phenomenological models. No.

No.

Comment

gi

B

isolated form with temperature

1

1

1

2

1

3

0

5

gi

B

6

1

10

0

11 12

See above

where or

6

[T/Tr]

−m

m

[(Tm − T)/(Tm − Tr)]

7 8 9-

or

or

or

.

, or

or

*

T

1 + λ(e

Ta/Tm

−e

e a m) 1 − D(T − Tr)

)/(e −

exp ( − D[(T − Tr)/(Tm − m

Johnson-Cook thermal softening model [6] Power law model [61]

Tr)] )

Khan model [50] Thermal softening model [62], suitable for HCP materials Linear model [63]

Thermal softening model exponential model 1 [64] Thermal softening model exponential model 2 [65] Thermal softening model exponential model 3 [66] Thermal softening model exponential model 4 [67]

The coupled strain-rate sensitivity function with temperature is chosen from Table 2 or any weighted combination of those strain-rate sensitivity functions. Step 10:

is verified for feasibility by performing a multi-objective fit of the trial constitutive model in Eqs. (13) or (14) to multiple dynamic curves of varying flow stress at several high plastic strain values with strain-rate at different forming temperatures. If a good fit is achieved, the basic form of constitutive model is formulated by the above identification procedures. Otherwise, go to Step 9 and revise it. (13)

or

;

(14)

Step 11: A trial full constitutive model , formulated using dependence-based integrated methodology, is fit to all the experimental data point using a weighted multi-objective identification strategy such that a set of material parameter values are identified. The model feasibility is validated by measuring the overall fit quality, average absolute relative error and asymptotic fit standard error. If a good overall quality of fit is available, the identification processes of constitutive model and its material parameters are finished. Otherwise, go to Steps 3, 5, 7 or 9 for the corresponding modification of strain hardening, strain-rate sensitivity or thermal softening functions, according to the relative fit standard parameter error RFSPE in Eq. (15), equal to the fit standard parameter relative errors FSPE divided by the parameter vector P, until the specified error criteria above are fulfilled simultaneously.

(11)

(12)

Step 9:

m

1 − [(T − Tr)/(Tm − Tr)]

T /T

5

rise), using multi-objective identification strategy. Otherwise, go to Step 7 and revise it.

or

Comment

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9

4

coupled form with temperature Variable strain rate sensitivity [56] Reduced rate strengthening at high rate [57] Enhanced rate strengthening model 1 [58] Coupled rate-temp ZA model for BCC materials [8] Coupled rate-temp exponential model [59] Coupled rate-temp power model [60] Coupling with temperature

7 8

3

Johnson-Cook rate dependent model [6] Cowper-Symonds model [52] Power law model [53] Wagoner model [54] Rate sensitivity exponential model [55] Comment

4

No.

2

h(T)

PR OO F

Table 2 Typical strain-rate sensitivity functions in common phenomenological models.

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but varying with plastic strain history, as vividly illustrated in Fig. 2. Thus, the artificial strengthening of thermal softening effect will lead to an inaccurate estimate of strain-rate sensitivity coefficient. Realistically, varying forming temperatures and strain rates with deformation (see the Appendix A) limits the identification of constitutive parameters using classical non-linear regression strategy. Since the nonconstant nominal/average strain rates and forming temperatures during process are considered in accurate identification of constitutive parameter, an improved and generalized parameter identification based on the weighted multi-objective methodology is proposed to find a more accurate set of material parameters over the entire span of quasi-static and dynamic modes.

(15)

PR OO F

where J and W are defined only after in the manuscript. 3. Weighted multi-objective strategy for material parameters identification 3.1. Identification of material parameters: background The commonly used methods for material parameter determination are divided into two categories: inverse analysis and non-linear regression. Inverse analysis is a minimization method of determining the material parameters by solving the least-square sum of the deviation between experimental data in a global range and the model prediction values at the same strain or by performing finite element analysis for comparable simulation results to experimental observation [68,69]. This non-linear regression is to obtain multiple sets of parameter values under localized experimental values in deformation history, through univariate analysis of constitutive model and then to average them to find a final set of parameters [26]. The nonlinear regression approach for parameter identification is implemented successively through the following routines:

3.2. Weighted multi-objective identification methodology

Together with the above dependence-based integrated methodology for constitutive identification, an improved multi-objective identification methodology for material parameters is proposed to arrive at a more accurate, reliable, and generalized set of material parameters. To do this, this paper is based on the criteria from Ref. [71] and suggests a modified version of some criteria for the formulation of objective function as follows.

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Criterion 1 The measurement errors of experimental data in one deformation mode should be taken into account in the formulation of objective function.

– The material parameters for work-hardening function can be evaluated through nonlinear regression from quasi-static test, i.e. at reference strain-rate, forming temperature, and other physical or mechanical properties. – The parameters for strain rate sensitivity can be identified by performing regression analysis upon mathematical transformation of dynamic test data in the reference temperature under different plastic strain, and averaging the inversely-transformed parameter values under different plastic strain to obtain final evaluation of strain rate sensitivity parameters. – The parameters for thermal softening can be estimated using the similar approach to the one for determining strain rate sensitivity parameters. – The parameters for characterizing the dependence of flow stress on other physical or mechanical processes, can be calculated by performing regression analysis of flow stress against this physical or mechanical variable under different levels of one independent variable such as strain, strain-rate, or temperature, and later performing nonlinear fit of the physical or mechanical variable against the independent variable. Its identification process is actually more complex due to the coupled effects of it and other independent variables.

Criterion 2 For single curve, all the experimental data points should be accounted for and have equal pressure to be optimized during parameters determination. When optimizing multiple curves corresponding to different deformation modes, it should be ensured that all experimental curves are endowed with equal optimization opportunities, with independence of the number of experimental data in any deformation mode. If involving multi-sub-objectives, individual objective function should be capable of dealing with incorporation of multiple sub-objectives by assigning equal opportunity to be optimized for individual sub-objective, independently of the number of data points in each sub-objective. Criterion 3 Continuity condition of multi-objective function should be fulfilled by means of progressive evaluation of fitting quality

The constant (average) strain-rate experimental data are employed for determining material parameters for strain-hardening and thermal softening functions through the conventional nonlinear regression approach. The isothermal uniaxial mechanical test is a commonly used method for determining the flow stress of a material. The quasi-static test is usually carried out on a universal testing machine. However, the motion is applied at constant travel speed (equal to the product of nominal strain rate and gauge length hw) rather than constant nominal strain-rate on the universal testing machine. In the case of large deformation, the deviation from nominal strain-rate have significant influence upon determination of strain-hardening model of the materials showing the sensitivity of flow stress on strain rate even in quasi-static mode. Moreover, the temperature rise during deformation is not constant

1 Fig. 2. Computed temperature rise of SiCp /Al composites from plastic strain at 6000 s− and 100 °C.

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since the gradient-based optimization algorithm cannot be executed in a discontinuous function.

give equal opportunity to each objective function corresponding to one loading mode, the normalization is performed on all objective functions by considering measurement error. Generally, the measure deviation of all experimental points in one deformation mode should be of either constant measurement error, or proportional error to flow stress . Under the assumption of constant measurement error in each loading mode, since the measurement error are unknown, the covariance analysis can be performed to arrive at the measurement error [72]. The measurement error in any deformation

PR OO F

Criterion 4 The weighting factors, used for allocating equal opportunity to be optimized for each curve, should be assigned automatically based on available statistical evaluations. A general methodology for inverse identification of material parameters P is to minimize the difference between experimentally observed material behavior and constitutive model predictions, with the following basic formalism.

mode is evaluated from the following fitting for the ith experimental data point in the jth deformation mode.

(16) s.t. (17)

where k is scaling factor. The weighting factors obtained by variance analysis of measurement errors are here introduced to formulate multi-objective function.

(18) where P is a vector of parameters to be fitted, Nj the number of exper-

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imental points under the jth loading condition, and M the number of loading conditions. The superscripts e and m label the experimental and constitutive-based calculated values, respectively. The accuracy and reliability of inverse analysis strategy for parameter identification are dependent upon the information involved in the objective function, which is arrived at by incorporation of weighting factors in the formulation of multi-objective function [71]. A proper definition of multi-objective function, thus, will result in more accurate identification for constitutive parameters. As can be seen in Fig. 3, the measurement deviation/error of flow stress obtained from dynamic experiments form mean curve at different forming temperatures might be of different scales, i.e. 21 MPa at low temperature and 8 MPa at high temperature within 99% confidence interval. It should be noted that if the measurement errors in different loading modes were of different orders of magnitude, a set of estimates for material parameters might minimize the specific objective function in one deformation mode with larger measurement error and not all objective functions. This would lead the fit material model to perform well in one loading mode but not in others. To

The assignment of weighting factors based on measurement error covariance to different deformation modes also contributes to satisfy the Criteria 1 and 4. Simultaneously, the influence of those individual objective functions with poor quality of fit on multi-objective minimization process is reduced by the assignment of this weighting factor, denoting individual fit standard error. More precisely, during multi-objective optimization process for parameter identification, it is also important to note that equal opportunities to be optimized is given to each objective function corresponding to one mechanical test. Otherwise, the fit constitutive model may behave well in predicting mechanical responses in one loading condition but not in others. This is very likely if the number of experimental points and/or number of active constitutive parameters (less parameters in quasi-static mode) in different loading conditions are of magnitude. The weighting factors in the jth deformation mode considering the number of experimental points and number of active constitutive parameters should be formulated in terms of Criteria 2 and 4, as follows. (21)

Therefore, coupled with the assignment of weighting factors in Eqs. (20) and (21), the multi-objective function in Eq. (16) can be reformulated by (22)

The above multi-objective function can be rearranged to be non-dimensionalized, taking the following relative formalism

Fig. 3. Schematic of flow stress measurement deviation at different forming temperatures.

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of

W is weighting matrix, of which the weighting component for the ith experimental data point in the jth deformation mode with respect to the kth material parameter can be formulated in the multiplicative form of weighting factors in Eqs. (20), (21) and (27) as

PR OO F

(23) Form the Eq. (23), it implies that this relative multi-objective function normalized for parameter identification can be used for concurrent optimization of other available material behaviors with different units. The sub-objective functions in quasi-static and dynamic modes can be derived from Eq. (22) as

J is Jacobian matrix, of which the component Jik is the partial deriv-

ative of the ith experimental point parameter Pk in P, as expressed by

(24)

with respect to the kth material

T

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may lead to the unachievable inversion of [J WJ + λI]. An alternative approach aiming at defining a suitable λ for Levenberg's algorithm was suggested by Nielsen [73]. It is proved that the Levenberg-Nielsen algorithm is of better performance on convergence and time cost, compared to Levenberg–Marquardt algorithm. An effective initial value of the damping factor λ0 is recommended as follows.

To be capable of representing the quasi-static and dynamic material behavior simultaneously and equally well, an equally weighted bio-objective function is formulated in an additive form of quasi-static and dynamic objective functions in Eqs. (24) and (25) as (26)

(32)

where τ is an initial guess for which the recommended value range is 6 to 10− 3, beyond which the poor guess is made [74]. from 10− If the metric , the damping factors are iteratively updated according to the following criteria,

(27)

and

(31)

3.3.2. Initialization and update of damping factor λ and step length h If the damping factor λ in Eq. (29) is large, Levenberg's algorithm

(25)

with

(30)

(33)

and if Q(hi) ≤ ϵ4, then

(28)

(34)

The weighted multi-objective optimization for inverse identification for constitutive parameters is carried out by the Levenberg algorithm combined with the suggested damping factor by Nielsen [73], with good performance on convergence and time cost.

where νi is scaling factor, Q(hi) is a ratio between the actual and expected improvement in the objective function [75].

3.3. Numerical implementation

(35)

ϵ4 is a specified threshold controlling the acceptance of step update

3.3.1. Algorithm The weighted multi-objective optimization for inverse identification for constitutive parameters is carried out by the Levenberg algorithm for solving the nonlinear least-square problem in Eq. (26), as follows.

h. If

, indicating Pi + hi is superior to Pi, Pi is replaced by

Pi + hi. Otherwise, the damping factor is updated according to Eq. (34), and the algorithm proceeds to the next iterative process.

(29)

3.3.3. Error analysis The coefficient of determination, R2 , is used for evaluating the overall quality of fit, which provides a direct measure of how well experimental observations are replicated by fitting model.

Here Pi + 1 = Pi + hi, where Pi and hi are the fit vector and step length vector of the parameters at the iteration step s,

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(36)

PR OO F

with (37) It is worth noting, additionally, that even a higher R2 may not necessarily imply better predictability and reliability since fitting model tends to be biased toward lower or higher evaluation [76]. Thus, the average absolute relative error AARE and asymptotic fit standard error AFSE are used to give an unbiased statistical error measure of model predictability and reliability, with the following formalism.

Fig. 4. Microstructure of Al6061/SiCp/15 composites.

height to perform temperature measure. The specimen was preheated at 4 K/s up to a desired temperature, and then held for 3 min at this forming temperature. The cylindrical samples of Φ6 mm × 9 mm in size were 1, at nominal forming temcompressed at constant strain rate of 0.001 s− peratures of 25, 100, 200, 300 and 400 °C in a vacuum chamber. The dynamic compressive tests were performed for the cylindrical specimen of Φ4 mm × 4 mm on a SHPB apparatus, at the nominal forming temperatures of 25,100, and 200 °C, at the nominal strain rates of 1. The SHPB experimental details are pre1000, 2000, 5000 and 7000 s− sented in Ref. [78].

(38)

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(39) The fit standard parameter errors FSPE, as a measure of experimental data variability effect on the parameters variability, can be calculated by taking the square root of the leading diagonal elements of the variance-covariance matrix of the parameter vector P.

4.3. Results

(40)

Fig. 5(a) and (b) respectively illustrate the quasi-static and dynamic true stress-strain curves of Al6061/SiCp /15 composites at different forming temperatures, with the offset yield point Rp 0.2 = 264.0 MPa. From quasi-static curves it is found that with varying forming temperature, the strain hardening behaviors show temperature dependence, and at high temperature than 400 °C, almost no strain hardening arise. The significant thermal softening effect of flow stress in quasi-static mode is temperature dependent, and so it is in dynamic loading mode. The strain-rate sensitivity of flow stress at identical forming temperature in dynamic mode are observed, especially at low forming temperature. Besides, the greater fluctuation of flow stress in dynamic mode than that in quasi-static mode may be ascribed to the nature of variable strain-rates in dynamic loading process.

From the methodology view of point, only by a fewer number of tests where both strain-rate and temperature can vary simultaneously, can the determination of material parameters be achieved using multi-objective identification strategy, compared to classical identification strategies. 4. Case study 4.1. Materials

SiC particulate reinforced Al matrix composites have received wide attention in aerospace, military, automobile, and civil manufacturing fields, due to high specific strength and stiffness, wear resistance, fatigue resistance, and tailored properties above, over conventional aluminum alloys [77]. The SiCp /Al composites employed for constitutive modelling is15% volume fraction SiC particulate reinforced Al 6061 matrix composites (Al6061/SiCp /15 composites) processed using pressure infiltration method. Using 3D Laser Scanning Microscope VK-X200, the micrograph of Al6061/SiCp/15 composites is illustrated in Fig. 4, showing the microstructure of SiC particulate with some clustering and no preferred orientation in Al matrix alloy. The reinforcement phase SiC particulates with many sharp corners are approximatively polyhedral and the average particulate size are about 5 µm.

4.4. Constitutive modelling Combined with the weighted multi-objective strategy for material parameters identification, the proposed dependence-based integrated methodology is employed for identifying constitutive model form and its material parameters for SiCp /Al composites. Single-objective fit of the suitable strain hardening functions chosen in Table 1 for Al6061/ SiCp /15 composites is performed for quasi-static mechanical data at 3 s− 1 and 25 °C. Meanwhile, an efficient set of loading conditions of 10− initial estimates are arrived at from a single-objective fit of the quasi-static data. Table 4 presents single-objective fit results of the suitable strain hardening functions. From a fit-quality view of point, the Shin, Polynomial, Jeong, Voce models can provide the capability of predicting strain hardening behavior in quasi-static mode well. Fig. 6 shows the experiment data and models prediction of flow stress-plastic strain of Al6061/ 3 − SiCp /15 composites at 10− s 1 and 25 °C. The prediction range of the

4.2. Experimental

The quasi-static compression experiments were conducted on Gleeble 3500 thermomechanical testing machine. The S-thermocouple was joined to the location near half the sample

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gives

with

PR OO F

(41)

(42)

where A is initial yield stress, B1 and B2 are strain-straining coefficients,

Wd is weighting coefficient, G is thermal softening coefficient, Tref and Tmelt are reference temperature and melting temperature, respectively.

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It yields a feasible fit results with the average absolute relative error of 6.6%, asymptotic fit standard error of 20.12 MPa, and R2 of 98.50%. The comparison of experimental data and identified model prediction is shown in Fig. 7(a). The overestimated flow stress by the identified model implies the temperature dependency of strain hardening behavior. Therefore, thermal softening effect is to be introduced to the mechanical characterization of strain hardening behavior. The chosen coupled strain hardening function form with temperature is not determined using weighted multi-objective identification method until it can be fit equally well to all quasi-static experimental curves, with an acceptable degree of fit. The multi-objective inverse identification yields a weighted thermal softening function form in Eq. (43), with (R2 , AARE, AFSE) being (99.70%, 4.30%, 7.7866 MPa). Comparison of experimental data and model prediction curves in quasi-static loading modes is presented in Fig. 7(b), illustrating the flexibility and accuracy of this weighted multi-objective identification strategy.

Fig. 5. True stress-plastic strain curves of Al6061/SiCp /15 composites at different forming temperatures (a) under quasi-static loading, and (b) under quasi-static loading.

Polynomial model is only within a narrow experimental plastic strain, and not extended to a large strain beyond the experimental data. The highest degree of fit for Jeong model is ascribed to more parameters to be fit, increasing the constitutive complexity. Voce and Shin models differ in equation form but coincide in nature. Hence, Voce or Shin models can be chosen for the function formalism describing strain hardening behavior of Al6061/SiCp /15 composites. The experimental data of flow stress-forming temperature at different plastic strains in quasi-static mode are chosen as many as possible for multi-objective identification of thermal softening function. The optimum form of thermal softening function

(43)

There exists a small deviation between experimental data and model prediction at 100 °C, and the experimental flow stress curve at 100 °C shows stronger strain hardening than

Table 4 Single-objective fit results of the suitable strain hardening functions. Model

Ludwik G-Ludwik Shin

Polynomial Jeong Voce

Fit equation

Overall quality of fit

0.1952

231.7 + 229.4εp

0.0000457 + 448.3(εp + 0.0007156)

0.07298

290.9 + 105.7[1 − exp( − 22.8εp)] −

4 1.7E5εp

+

3 1.04E4εp

400.8 − 125.9exp( −

−

2 2.33E4εp

0.7306 10.78εp )

R2

AARE

AFSE

σ(εp = 0)

95.14%

2.072%

5.661

231.7

97.51%

1.350%

4.055

264.2

99.30%

0.631%

2.146

290.9

99.92%

0.177%

0.707

274.9

99.30%

0.631%

2.146

290.9

99.51%

+ 2397εp + 290

396.5 − 105.6exp( − 22.8εp)

10

0.487%

1.797

290.0

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PR OO F

objective inverse identification strategy, with the fit results illustrated in Fig. 7(d). A best fit to the above experimental data with a high fit quality of R2 = 98.94%, AARE= 1.38%, and AFSE = 4.74 MPa, is identified as (45)

Thus, a trial constitutive model for predicting material behaviors of Al6061/SiCp /15 composites in static and dynamic modes is established in Eq. (46). Since the above identification process of material constitutive formalism is aimed at the fit to local experimental data but not all ones, the final objective is to find a set of material parameters arrived at using multi-objective inverse identification strategy by fitting the identified constitutive model well to all the experimental data.

(46)

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The quantitative identification results of fit upon R2 , average absolute relative error and asymptotic fit standard error are found to be 99.62%, 2.34%, and 8.9138 MPa, respectively. The material parameter of the identified constitutive model above for Al6061/SiCp /15 composites are presented in Table 5. The experimental flow stress-plastic strain curves in both quasi-static and dynamic loading modes are illustrated in Fig. 8, together with the corresponding simulations using the identified constitutive model. A good fit indicates that the proposed dependence-based integrated constitutive modelling methodology, together with multi-objective parameter identification strategy may provide a new methodology in characterizing material behaviors in quasi-static and dynamic modes equally well.

3 s− 1 and Fig. 6. Experiment and models prediction of flow stress-plastic strain at 10− 25 °C.

those at other temperatures. This special phenomenon may be due to the increase in the diffusion rate in this temperature range from 100 °C to 400 °C that make the diffusion become active. In order to reduce the temperature dependencies in mechanical characterization of strain-rate sensitivity behavior, the variation of flow stress at very small plastic strain of 0%, 0.2%, 0.5% and 1% with strain rate under reference temperature 25 °C are employed for identifying isolated strain-rate sensitivity function through multi-objective fit strategy. Fig. 7(c) illustrates the experimental data and multi-objective fit curves in dynamic loading modes. When the Cowper-Symonds rate-dependent function in Table 2 is applied in Eq. (44), it yields a better fit quality of R2 = 94.0%, AARE= 2.52%, and AFSE = 7.74 MPa, over other strain-rate sensitivity ones. Fig. 7(c) shows the experimental and multi-objective fit curves at small plastic strains in dynamic loading modes.

5. Conclusions

The phenomenological model is widely used for the characterization of plastic behavior of metals and alloys during forming and manufacturing. But the applications of phenomenological constitutive model in industrial simulation are often faced with some major difficulties such as the requirement of a larger number of mechanical tests over a wide loading range by the classical parameter identification strategy, determination of coupled relations among explanatory variables, and the ensuing hard identification job of material parameters involved in constitutive characterization. This paper suggests an improved methodology for dependence-based integrated development of constitutive model, together with multi-objective identification scheme of material parameters with the equal capability of predicting material behaviors in quasi-static and dynamic modes well. The automatic assignment of three weighting factors considering (a) different measure errors or individual fit standard errors in individual objective function corresponding to each test, (b) the number of experimental points and number of active material parameters under different loading conditions, and (c) assignment of equal optimization opportunity for quasi-static and dynamic sub-objective functions, is introduced in the formulation of multi-objective identification model to arrive at a more accurate, reliable, and generalized set of material parameters over the entire span of

(44)

As can be seen from the overestimated flow stress in Fig. 7(c) and a not-too-high R2 value, the effect of temperature dependency is to be incorporated into strain-rate sensitivity function to arrive at a high degree of fit. Hence the ensuing identification task is to find the suitable strain-rate sensitivity function with temperature dependencies for capturing general tendency of strain rate-dependence. Multiple dynamic curves of varying flow stress at higher plastic strain values of 10%, 15%, and 20% with strain-rate at forming temperatures of 25 °C, 100 °C and 200 °C are employed to identify specific coupled relation of strain rate hardening with temperature using multi-

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Fig. 7. Experimental and multi-objective fit curves (a) at different plastic strains within a large range, (b) in quasi-static loading modes, (c) at small plastic strains at 25 °C, and (d) at larger plastic strains under different temperatures. Table 5 Multi-objective fit results of the identified full model for Al6061/SiCp /15 composites. Para. Value FSPE

A

B1

B2

n1

Wd

n3

n2

D

m

G

314.2 1.306

74.45 1.245

24.84 0.787

1.387 0.105

0.8584 0.003

4.652 0.044

2.385 0.081

1,034,000 191,850

4.389 0.164

72.82 3.224

quasi-static and dynamic loading conditions, independently of a user. The performance of this dependence-based integrated constitutive modelling methodology and multi-objective parameter identification strategy is verified by illustrating the multi-objective fit to the experimental data of Al6061/SiCp /15 composites from quasi-static Gleeble and dynamic SHPB tests with the instantaneous strain rates variation and corrected plastic strain-related forming temperatures changes, where the classical parameter identification strategy cannot work. A good fit shows the feasibility of the proposed dependence-based integrated constitutive methodology and the qualification of weighted multi-objective identification strategy in characterizing overall material behaviors in both quasi-static and dynamic modes.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51575051) and National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2012ZX04003051-3). The authors are also grateful for the technical support from the School of Aeronautics at Northwestern Polytechnical University. Conflict of interest The authors declare no competing financial interests.

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Appendix A

PR OO F

Though the influence of the variation of strain rate on most engineering materials in quasi-static test is not considerably significant, it is of significant importance to incorporate strain rate effect into a correct determination of constitutive parameters in quasi-static mode. Hence under kinematic conditions of large displacements and large strains, the true strain rate for the ith experimental data point can be corrected from Eq. (A1). (A1)

where di is the displacement for the ith experimental data point. But

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if so, the classical identification strategy cannot be used in parameter identification of strain hardening function. In the classical identification strategy, only a constant (average) nominal strain rate, instead of practical strain rate corresponding to each data point in dynamic test, is employed during parameter identification process. To improve the fit precision of constitutive model, instantaneous strain rates in dynamic tests are chosen for optimization. In this case, the conventional nonlinear regression approach is limited due to nonconstant strain rate. An alternative quasi-static test without any correction is Gleeble thermomechanical testing, where the instantaneous strain rate can be kept nearly constant. It is worth noting that in isothermal uniaxial test when strain rate 3 s− 1, the heat generated by plastic deformation are dissipated < 10− timely due to sufficient deformation time, so the test at strain rate of 3 s− 1 can be taken as isothermal deformation. When strain less than 10− 3 s− 1, thermal-softening effect during deformation is rate exceeds 10− so considerable that non-isothermal deformation with temperature correction should be taken into account [70]. As the strain rate increases 1, the plastic work during deformation results in conversion of a to 1 s− considerable fraction of plastic strain energy into adiabatic heat in localized regime. The quantitative relation between plastic work ∆W and converted heat including dissipated heat can be given based on the first law of thermodynamics, as follows. (A2)

Here ∆W = ∫σdεp and Qw = ρCp∆T, where ρ and Cp are respectively

density and specific heat, Δεp plastic strain during deformation. η is the

Taylor–Quinney coefficient that stands for the fraction of conversion of mechanical work into heat, typically assumed to be 0.9–0.95, whereas the remaining fraction used for microstructure change (grain evolution, dynamic recrystallization, phase transformation, etc.). The adiabatic correction factor α, represents the fraction of adiabatic heat remaining in test specimen due to heat dissipation into die and environment. In terms of strain rate range, the adiabatic correction factor varying with strain rate is defined as [70]

Fig. 8. Comparison of experimental and simulations using multi-objective identification constitutive model in the modes of (a) quasi-static loading, (b) dynamic loading at 25 °C, and (c) dynamic loading at elevated temperatures.

(A3)

with

Author contributions

J.F.X. and L.J.X. designed this research. J.F.X.and F.N.G. carried out quasi-static and dynamic mechanical tests. J.F.X. wrote the Matlab codes of multi-objective identification strategy for material parameters. J.F.X., L.J.X., J.Y., S.Q.P. and X.B. Wang wrote this manuscript.

(A4)

where hw is the workpiece height, dd the distance from die surface to die interior where the temperature keeps constant, κinterface heat transfer

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coefficient

of

work/die

interface,

κw

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and κd thermal conductivity of workpiece and die, respectively. From

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