Inverse Modeling Applied for Material Characterization of Powder Materials

Inverse Modeling Applied for Material Characterization of Powder Materials

Article  in  Journal of Testing and Evaluation · September 2015

DOI: 10.1520/JTE20130266

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Testing and Evaluation

Daniel C. Andersson,

¹

Per Lindskog,

²

and Per-Lennart Larsson

³

DOI: 10.1520/JTE20130266

Inverse Modeling Applied for Material Characterization of Powder Materials

VOL. 43 / NO. 5 / SEPTEMBER 2015

Daniel C. Andersson,¹Per Lindskog,²and Per-Lennart Larsson³

Inverse Modeling Applied for Material Characterization of Powder Materials

Reference

Andersson, Daniel C., Lindskog, Per, and Larsson, Per-Lennart, “Inverse Modeling Applied for Material Characterization of Powder Materials,”Journal of Testing and Evaluation, Vol. 43, No. 5, 2015, pp. 1005–

1019, doi:10.1520/JTE20130266. ISSN 0090-3973

ABSTRACT

An investigation is performed concerning the applicability of inverse procedures, using optimization and simple experiments, for characterization of WC/Co powder materials. The numerical procedure is combined with uniaxial die-compaction experiments using an instrumented die, which allows direct measurement of the distribution of radial stress during the experiments. Finite-element (FE) methods and an advanced constitutive description of powder materials are relied upon to model the compaction experiment. Optimization using a surrogate model is used to determine some of the parameters in the constitutive description. These parameters in the material model are said to be found (with some accuracy) if the output from the FE simulation is similar to the experimental data. It is found that even though a complete constitutive description of the powder materials investigated cannot be achieved using this approach, many important material parameters can be determined with good accuracy.

Keywords

powder compaction, constitutive description, inverse modeling, optimization, parameter sensitivity, material characterization

Introduction

To put a new product on the market, being a cemented carbide tool manufacturer, the experimen- tal trial and error process of getting the right powder into the right product geometry is often time consuming and thus, expensive. This involves iteration of tooling geometries, powder mixtures and press and sintering parameters. To facilitate this, one approach is to increase the time spent on simulation of the process and less on experimental work, potentially speeding up the product

Manuscript received October 11, 2013;

accepted for publication June 3, 2014;

published online October 10, 2014.

1Dept. of Solid Mechanics, Royal Institute of Technology, SE-10044 Stockholm, Sweden.

2Sandvik Coromant AB, SE-12680 Stockholm, Sweden.

3Dept. of Solid Mechanics, Royal Institute of Technology, SE-10044 Stockholm, Sweden (Corresponding author), e-mail: pelle@hallf.kth.se

CopyrightVC2014 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. 1005 doi:10.1520/JTE20130266 / Vol. 43 / No. 5 / September 2015 / available online at www.astm.org

development. In this case, the precision of the numerical tools are important including a good mechanical description of the material.

In this study, the die-compaction process, from now on of- ten referred to as simply powder compaction, is studied. It is the test most frequently used in forming of powder materials today, and it is used across many different industries and for many different applications. The process of performing such a test, involves filling of the die, pressing of the powder in the die, and ejection of the powder from the die. During the compaction process, whether it is die compaction or other types of loading, there will be a non-uniform distribution of density in the com- pressed powder (e.g., green body), which will introduce shape changes in the sintered body. The goal when studying compac- tion is to be capable to predict the density distribution, and other mechanically related features, such as cracks, in the green bodies. Three stages in the test can be distinguished. Obviously then, the filling of the die with powder material is the first stage.

Second, pressing up to a low density is performed and, modeled either analytically, cf., e.g., Refs1–4, or using numerical meth- ods, like the discrete-element method, cf., e.g., Refs5–10. These methods are based on carefully deduced compliance relations, cf., e.g., Refs11–14. Finally, pressing up to high densities (and ejection of the green body from the die after unloading) is per- formed. In this situation the material is often treated as a porous solid and phenomenologically modeled using well-known mate- rial models, like the Gurson model [15], the CamClay model [16], or some type of Drucker-Prager CAP model [17]. To com- pletely evaluate a compaction test, sintering of the pressed- powder green body is, of course, necessary to understand in detail. The sintering step is, however, not included in this investigation but is left for future studies.

In many macroscopic constitutive models aiming at deter- mining the mechanical behavior during compaction, the elastic behavior is usually described by a density dependent bulk mod- ulus (or Young’s modulus), and Poisson’s ratio. These proper- ties can be determined by hydrostatic tests, as well as uniaxial compression tests, cf. Ref 18. Traditionally, determining the plastic properties of the powder compact in compaction is done using sophisticated triaxial testing equipment (which of course also can be used to determine the elastic properties of the pow- der). Basically it can be concluded though, that when very com- plicated constitutive models of powder compaction are at issue, as the one presented in Ref 19, the experimental burden of determining all of the parameters is substantial. The model in Ref 19, which includes many constitutive parameters, will be studied in detail in the present analysis. In such a case, several different testing machines and experimental setups are needed, as well as expensive, time-consuming laboratory time/work.

As an alternative to spending large efforts on different types of mechanical testing, inverse methods, as described in Ref20, are often considered. The basic idea is that more or less involved

testing configurations, cf., e.g., Ref21, can contain a lot of infor- mation, and if this kind of test is combined with inverse meth- ods, for example, optimization, the intricate relationships between the parameters in the system studied can possibly be understood in a better way than simply by intuition. At material characterization, as in the present study, a considerable amount of pertinent work has been presented, cf., e.g., Refs22–24, and now inverse methods for this purpose is considered almost a standard procedure for many materials. When the material behavior is complicated, as in the present case with hard metal powders at compaction, and, thus, the corresponding constitu- tive model contains many parameters, cf., e.g., Ref19, progress is more limited. This problem will be considered in this study.

In doing so, the present approach is to use the geometri- cally simple die pressing experiment, attempting to increase the amount of information normally extracted from such an experi- ment by using sophisticated measuring equipment, and then applying inverse modeling to determine the relevant material parameters. This has, for less complex material models than the one considered presently, been investigated before, in, for exam- ple, Refs 24 and25. In the work by Wikman et al. [24], iron powder was compacted using a single action pressing in a die with varying radius in the press direction, and inverse modeling was applied using a modified Nelder-Mead simplex method to determine parameters in the material model. The material model used was a two-invariant, rate-independent elastoplastic CAP model with an associated flow rule based on the classical Drucker-Prager CAP model [15]. Hrairi et al. [25] also studied iron powder combining finite-element methods (FEM), inverse modeling and, as in Ref24, relying on a material model based on the Drucker-Prager CAP model [17]. In summary, progress was reported in both Refs 24and25, but a complete material characterization was not achieved.

The present investigation utilizes an instrumented die, described in detail in Ref 26, with a cylindrical cavity having constant radius and with eight pressure sensors in the die placed in a helical pattern around the powder to measure, directly on the powder surface, the distribution of the radial pressure from the powder onto the wall of the die. It is the aim of the present study to determine whether this additional information is suffi- cient to determine relevant material characteristics when using an advanced constitutive model with many constitutive quantities.

Procedure

In the present study, experiments using the instrumented die, numerical analysis using finite elements, and inverse modeling using optimization are combined with advantages taken of com- mercial software combined with additional external programs.

The validation of the procedure is done by using the constitu- tive model [19] relied upon throughout the study to produce a

reference set of data (which in the real case would be experi- mental data from an instrumented die-compaction experiment) from a numerical simulation using finite elements. The inverse method is then set up to determine the model parameters in the reference case (by trying to reproduce the reference data). If the agreement between the reference data (press and friction forces as well as the radial pressure distribution) and inverse solution is good, meaning that the model parameter values obtained is the same as in the reference case, then it is argued that the method is promising.

It is not expected that all of the parameters in the constitu- tive model [19] used presently can be determined from the pres- ent procedure. Many features exhibited by hard metal powder materials may not affect the measured output from such a rela- tively simple geometrical case as the cylindrical die-compaction experiment. For that reason, a sensitivity analysis is performed on all of the parameters in the constitutive description to deter- mine which parameters are in fact possible to determine. The outcome from the sensitivity analysis is a set of parameters, which is deemed impossible to determine with this approach, and a set of parameters, which is said to be possible to deter- mine with some accuracy. Then, as described above, an attempt to determine the relevant (given by the sensitivity analysis) parameters in the constitutive model for a case with simulated experimental data is performed. If successful, then the same inverse analysis is performed using real experimental data perti- nent to two slightly different powder materials. It should here be clearly stated that the parameters, which, according to the sensitivity analysis, are not possible to determine with sufficient accuracy. These parameters are assigned values based on other experiments and for powders slightly different from the ones used in this investigation. The values determined for the param- eters included in the optimization will of course be affected by this, and they should therefore be seen as “best fit values” for the experimental data.

Andersson et al. [27] studied the influence from punch geometry on the stress distribution in the powder compact, and the conclusion from this study was that the influence is not very significant but that a skewed upper punch, if any, was prefera- ble. To determine whether this feature can be used to advantage at the characterization of powder materials at compaction, a sensitivity analysis using simulated reference data with a skewed upper punch is performed. These results are then compared to the corresponding ones pertinent to optimization with a flat upper punch.

In summary, the process of performing inverse analysis to determine material model parameters requires a number of things. First, an experimental procedure that activates the fea- tures of the powder material represented by the material model parameters must be determined. Second, reliable experimental data is of course crucial. Third, a numerical (FEM) model of the experiment, which in a relevant way represents the physical

experiment, is needed. Finally, a robust inverse procedure must be applied. It is important to stress that this investigation does not claim to develop optimization methods, but simply to apply existing methods available in commercial software in a practical manner.

Press Experiment

It should be clearly stated that all of the die-compaction experi- mental data used in the present study are based on the experi- mental findings in Ref26. These experiments are described in detail below. The uniaxial die-compaction experiment is shown inFig.1. The die is manufactured from hard metal, with a steel ring around, and is much stiffer than the powder material. The deformation of the die is purely elastic. The specification of the die is shown in^Tables1and², and the material constants are pertinent to a hypoelastic material. The upper punch is allowed to move during compaction, whereas the lower one remains fixed. It goes without saying that frictional contact will occur between tooling (die and punches) and powder material. It should be noted in passing that in the numerical part of the investigation (see below), this is modeled using standard Coulomb friction according to

s¼lp

(1)

where:

s¼the shear stress,

p¼the pressure directed normal to the contact surface, and

FIG. 1 The finite-element mesh used in all the calculations. The sensors positioned in a helical pattern around the powder are also indicated.

l¼the coefficient of friction.

Based on experimental results in Ref 28, for hard metal powders,

l¼0:2

(2)

is considered to be a fairly good assumption at compaction of cemented carbides. This value on lwill be used in all of the numerical calculations. The contact pressure between the die and the powder material is continuously recorded along the die wall using eight pressure sensors. The pressure sensors are indi- cated inFig.1and are positioned in a helical pattern around the powder, with the distance in the press direction between two sensors being 2.5 mm, and the angular distance 45.

Addressing the question of activation of material mecha- nisms and model parameters, the uniaxial die-compaction experiment produce a non-homogeneous stress state in the powder compact because of frictional effects, as shown in Ref 27. Also, the ratio between the hydrostatic stress and the deviatoric stress varies along the height of the powder compact during compaction. This indicates that the measured contact pressure along the height should reflect different points on the yield surface dependent on which sensor is studied.

The initial values on relative material density and height of the powder cylinder are determined from the position of the press, weight of the powder compact after pressing, and the radius of the die cavity. The deformations during pressing are then determined from the press piston displacement (given by the press) while adjusting for the compliance in the experiment.

The compliance of the upper punch and the compliance of the rig are accounted for separately, and they are approximated with non-linear functions. The amount of powder used in the experiment is calculated to have a final average relative density (actual density/sintered density) of 0.55 given a certain press cycle. After pressing, the green body is weighed and this value of the mass is used in the calculation of the density during pressing. The relative difference between the mass measured before and the mass measured after pressing is less than 0.1 %, which corresponds to a difference in radial pressure of around 1 %: this is also true for the press force. The experimental

procedure (and relevant results) is discussed in much more detail in Ref 26, and, indeed, it is the experimental outcome from this investigation that forms the basis for the inverse anal- ysis in the present study.

In summary, the information gathered from the experi- ments in Ref26and subsequently used in this investigation con- sists of press and friction forces, representing the global behavior of the powder compact, whereas the radial stress dis- tribution on the surface of the powder gives information about the local state. Despite the substantial amount of information gathered, it is not expected that (as indicated above) a complete constitutive description can be achieved using the present pro- cedure. This means that other experimental approaches will be needed for this purpose, and, as mentioned earlier, the present aim is to reduce the amount of such additional techniques.

Material Description and FE Analysis

Below, the presently used constitutive description, the powder materials used in the experiments, and the FEM simulations of the die-compaction experiments, are described in some detail.

CONSTITUTIVE DESCRIPTION

Hard metal powder exhibits a very complex material behavior at compaction, and to model such a behavior using a contin- uum approach, advanced constitutive descriptions are required, cf. Ref29. In this investigation, as mentioned repeatedly above, an experimentally verified model developed by Brandt and Nils- son [19] is used with parameter dependence and notation according to Ref27. These parameters are all listed in Refs19, 26, and27, and are also discussed below.

The yield surface (see Fig.2) in the material model is built up from four independent parameters, and a fifth, which is a function of the other four. The four independent parameters are the yield stress in hydrostatic compressionX(d), the yield stress in uniaxial compressionY(d), the aspect ratio of the elliptic cap R, and the constant ratioL/X, the intersection point of the fail- ure and cap surfaces. The fifth parameter is the yield stress in pure shear, i.e., the intersection point of the yield surface and the deviatoric axisc₀(d). The material model rests on the elasto- plastic theory with a yield surface resembling a Drucker-Prager CAP [17] surface. The yield surface is determined from a quad- ratic failure curve (yield function):

f^failureðrI;rII;dÞ ¼ ffiffiffiffiffiffirII

p ½c0ð Þ d c1ð Þrd Iþc2r²_I ¼0

(3)

and a standard elliptic cap part:

f^capðrI;rII;dÞ ¼ ffiffiffiffiffiffirII

p 1 R½ðL dð Þ

X dð ÞÞ²ðL dð Þ rIÞ²¹⁼²¼0

(4) TABLE 1 Material constants describing the hypoelastic material

behavior of the die used in the experiments.

Parameter Value Unit

Elastic Poisson’s ratio 0.22

Elastic modulus 580 GPa

TABLE 2 Explicit values on the geometrical dimensions of the elastic die used in the experiments.

Die inner radius 5 mm

Die outer radius 49 mm

Die height 50 mm

In Eqs3and4,

rI¼rii (5)

is the first invariant of the Cauchy stress tensor, and

rII¼1

2ðsiisjjsijsijÞ

(6)

is the second invariant of the deviatoric stress tensor s_ij. Furthermore,

c0ð Þ ¼ Yd ðdÞ LðdÞ² ffiffiffi3

p LðdÞ XðdÞ

R ðY dð Þ 2LðdÞÞ

½LðdÞ² þYðdÞðY dð Þ 2LðdÞÞ

(7a)

c2ð Þ ¼ d L dð Þ X dð Þ R Y dð Þ

ffiffiffi3 p

=½L dð Þ²þYðdÞðY dð Þ 2L dð ÞÞ

(7b)

c1ð Þ ¼d 2L dð Þc2ðdÞ

(7c)

with the material parameters above. At the intersection of the failure and cap curves, the condition:

@ð ffiffiffiffiffiffirII

p f^failureðrI;rII;dÞÞ

@rI

¼@ð ffiffiffiffiffiffirII

p f^capðrI;rII;dÞÞ

@rI (8)

is fulfilled.

The flow direction is determined from a non-associated flow rule where the flow angle offset (from a standard

“associated direction”), u, is determined experimentally as function of the parameter

Ju¼rii=XðdÞ

(9)

representing the proportion of the volumetric and the deviatoric part of the stress field.

Experimental data, discussed in Ref19, show that anisot- ropy develops during the compaction process and this is accounted for in the model through kinematic hardening with a back stressj. In the yield function a stress quantityris then replaced as r!r –j. Brandt and Nilsson [19] introduce a second-order tensorP, which is the stretch of deformation from the current relaxed configuration to an imaginary fully dense configuration. The back stress is coaxial with the deviator ofP according to

j¼hðeÞrI

P⁰ ffiffiffiffiffiffi P⁰_II p

(10)

where:

P⁰andP⁰_II¼the deviator and second invariant of the devia- tor ofP, and

h(e)¼an experimentally determined back stress scaling function.

The parametereis a scalar measure of the intensity of plas- tic anisotropy. For an isotropic powder and presently at initial

compaction,e¼1. This parameter is then updated based on the evolution of plastic shear strain, deviatoric plastic work, and volumetric plastic work in a linear manner using the coefficients c_a,c_g, andc_n(in^Table4), respectively.

CEMENTED CARBIDE POWDER MATERIAL

The materials used during the experiments are two slightly dif- ferent tungsten carbide (WC/Co) hard metal powders. The average WC particle size of one of the powder materials (mate- rial B) is approximately two to three times larger compared to the other (material A), which typically lead to lower press forces to achieve a certain relative material density. Furthermore, ma- terial A has two to five times larger granules than material B. In summary though, material B is expected to be easier to press, which means less compressive force for a given relative density, because of the larger WC particle size.

The material information base for this study consists of a large amount of experimental data published in Refs26,28, and 30, with this data translated into material model parameters in a constitutive model according to Refs19,31, and32. The data is valid for a powder material similar, but not identical, to the powders used here and because of this the material parameters for the presently investigated powders will be somewhat differ- ent. The average density range of experimental data used herein is between 0.45 and 0.60. The main reason for not using the whole set of experimental data is that the continuum approach is not valid in the lower density spectrum where the deforma- tion is governed by the contact interaction between powder par- ticles. Assuming that in the lower density spectrum the material density is (very) approximately half of the particle volume den- sity, this would indicate that the continuum approach is not valid ford<0.4, cf., e.g., Ref3. The higher limit is set because the pressing stage in production is limited to average relative densities of 0.55. The explicit experimental data used in the inverse modeling of material A is shown in^Fig.3(a)and^3(b) and correspondingly for material B inFig.3(c)and3(d).

PRESS SIMULATION

The quasi-static simulation of the pressing step is performed using FEM relying on the commercial code LS-DYNA [33]. The simulation approach has been discussed extensively in Refs26 and27. The experimental setup according to above is modeled assuming rigid tooling, hard contact, and Coulomb friction between tooling and powder. The finite-element mesh consists of 2106 deformable solid elements modeling the powder, with one integration point per element, and the model is shown in

Fig.1. To keep the integrity of the elements, the well-known arbitrary Lagrange-Euler (ALE) re-meshing method is utilized.

The loading is applied by prescribing a displacement on the remote boundary of the upper indenting punch, while keeping the lower punch fixed.

In the experimental setup, flat upper and lower indenters are used in all of the experiments. Because there was a small but noticeable difference in sensitivity between using a flat and skewed indenter with regard to the radial stress on the surface of the powder compact, cf. Ref27, a skewed upper indenter is also studied numerically.

Inverse Modeling

The objective function in the present study is a sum of the mean square error from 10 responses from the FE-simulation

and the corresponding experimental data. These are referred to as sub-objective functions and related to the press force applied to the upper surface of the powder, the total friction force between the powder and the die, and the eight radial stress quantities measured directly on the powder surface. The inverse modeling is controlled using the commercial software LS-OPT [34], with some assistance from external programs. The external programs are produced in the present study for the purpose of (1) calculating the normal pressure at sensor positions, (2) gen- erating material parameter functions from variable values, and (3) extracting the relative density values from the LS-DYNA [33] results. Variables here and throughout the study are what the optimization method handles, whereas parameters are perti- nent to the constitutive model or friction. In the case of constant material model parameters, the variable corresponding to the parameter takes on the value of the parameter, whereas in the case of a material model parameter dependent on the rel- ative material density, possibly more than one variable is used to represent this parameter. For example, the yield point in hydrostatic compressionX(d) is described using two variables, c01andc02, according to the Gaussian function

X dð Þ ¼ c01e

dkX

c₀₂

2

(11)

FIG. 3

Experimental data for material A and B taken from [26]. (a) and (c): The press and friction forces, defined in the text, pertinent to materials A and B, respectively, are shown as functions of the uniaxial compression of the powder (the mutual approach of the upper and lower punches).

(b) and (d): The radial stress distribution, determined by the pressure sensors, pertinent to materials A and B, respectively, are shown as function of the uniaxial compression of the powder (the mutual approach of the upper and lower punches).

FIG. 2 Schematic of the yield surface in the material model by Brandt and Nilsson [19] used throughout the numerical analysis.

where:

d¼the relative density, and

kX¼a constant given an explicit value prior to starting the optimization.

The coefficients c01 andc02 are the variables to be deter- mined in the optimization routine, and k_X is a constant set based on physical reasoning. The constantkXis set to a value that can be used for powders with different characteristics (the values in this paper are only tested for the two hard metal pow- ders used in the experiments). The actual value is set simply by plotting the resulting material parameter to make sure that the values over the density range of interest is physically reasonable.

The values of the two variables,c01andc02, are sensitive to the value of this constant, but within a limited range the corre- sponding variations ofc01andc02are sufficient to determine the functionX(d) with some flexibility.

In AppendixA, a detailed description of the combined use of LS-DYNA [33] and LS-OPT [34] for solving the inverse modeling problem is found.

SENSITIVITY ANALYSIS

The result from the sensitivity analysis is meant to indicate which material model parameters affect the output sufficiently for a possible determination when using the information gathered from the experiments described previously. Parameter values out of reach from the present procedure will of course introduce errors in the analysis, but this effect is hopefully small as these parameters are assigned values based on experimental data perti- nent to similar powder materials. In a practical situation, these (unknown) material parameters in the constitutive equations should be determined first using other types of experiments.

The sensitivity analysis is performed using the Sobol index [35]. The Sobol index represents a so-called higher-order sensi- tivity index, claiming to take into account not only the effect of one variable in the objective function, but considering also that varying this and the other variables in the system may affect the influence from this variable. A surrogate model is computed using second-degree polynomials [36], and the sensitivity indi- ces are computed from this surrogate model. For these indices to be relevant for the system, the surrogate model needs to be able to describe the actual system with some accuracy. In Ref36, radial basis functions and quadratic polynomials, Krig- ing, and multivariate adaptive regression splines (MARS) are compared, and, in most cases, the RBF model is recommended.

However, when dealing with problems that are only slightly non-linear in the parameter space studied, the second-degree polynomial is the most accurate and noise insensitive. In the present sensitivity studies, the RBF-description of the surrogate model is chosen because accuracy over the whole variable space is desirable. When trying to find the optimum values of the var- iables in the subsequent optimization, the linear or second- degree polynomial is recommended. In this case, second-order

polynomials are used, because compared to linear polynomials it has, in particular in this study, proven to be more robust.

OPTIMIZATION

The procedure to find the optimum values on the variables selected as a result of the sensitivity studies above is also con- trolled using LS-OPT [34]. A surrogate model constructed from linear polynomials is used, because the gradient of the target function with respect to the different variables are needed to find a very limited region where the optimum is located. A com- bination of a genetic algorithm (GA) [37] and a gradient-based method (LFOPC) [38] is used. The GA is a global search method that is intended to quickly find a limited region of inter- est in which the global optimum is located, whereas the gradient method is used to find the optimum in this region. The selec- tion of numerical experiments for the optimization is based on the D-optimality criterion. The boundaries of the variable space are then set based on physical reasoning. In doing so, possibly some extreme variable combinations cause termination because of unphysical material parameter values (like sign changes in parameters or similar). The experimental data used for identifi- cation of variables has been reduced to lie between an average relative material density of 0.45 and 0.55. The reason for this is, of course, as mentioned earlier, that low values of the relative density describe a powder that is difficult to model as a contin- uum (porous solid). Earlier relative material density data ranged from 0.42 to 0.66, and the experimental data from the experi- ments relied upon in this study covers average relative densities between 0.32 and 0.55. An average relative material density of 0.32 corresponds to an applied pressure of around 10 MPa in the experiments presented in Ref26.

The analysis of material A is done in such a way that the starting values on the material parameters are selected based on results from previously performed experiments, as discussed earlier, and thereby setting the boundaries on the values of the variables keeping the material parameters physically reasonable.

For material B, on the other hand, the starting values are set to the final values for material A. This is based on the fact that materials A and B are similar and their respective responses should then of course also be similar. The boundaries on the variable values for material B are then set tighter than for mate- rial A, as the resulting values are expected to lie close to the starting point. Another reason for keeping the variable bounda- ries more narrow for material B is that the risk of finding a local minimum is reduced, given that the optimum solution in the analysis of material A is a global optimum.

Results and Discussion

In this section, the qualities of the methodology used and the corresponding results produced are presented and discussed.

Basically, the result specifies a set of material parameters that

can be determined using the present approach and another set that cannot be determined. An appropriate outline then is to first specify the parameters that are possible to determine using this approach, and then to investigate the performance of the procedure when “simulated” experiments are studied. After the analysis using “simulated experimental data,” the real experi- ments are considered. The analysis of the powder materials with actual experimental data involves unknown material parame- ters, and when trying to match the experimental data using only some of them, the values of these parameters will have to com- pensate for the discrepancy in the values of the unknown parameters. Accordingly, the values will not only correspond to the physical meaning of the parameter, but also to some extent to the error in the values of the unknown parameters. If the method works, meaning that it determines the right values for the parameters, the values of the parameters will not be the same as those presented by Brandt and Nilsson [19]; but they will be quite close, as the presently used powder materials are slightly different as discussed previously. In this context, it should also be emphasized that the material model used in this investigation is a slight simplification of the model used in Ref19, as described in detail in Ref31, and also below.

Based on the microstructure of the powder materials A and B, their respective pressing characteristics should be somewhat different. In short, the size of the hard particles is smaller for material A, and, accordingly, the press force required to reach a certain average density should be higher. As seen in Fig.3, the experimental values pertinent to material A and material B are close, regarding both press forces and radial stress distribution, but not identical.

INITIAL DENSITY AND FRICTION

The results presented in this investigation are often shown explicitly or implicitly as a function of the material density of the powder. It is then of course necessary to accurately determine the initial density of the powder at the start of compaction, as both global and local variables are very sensitive to this quantity. The procedure of determining the initial volume of the powder is detailed in Ref26, and also the measurement of the weight of the powder. It should be emphasized that if the uniaxial compression of the powder is used to present the results in the figures below, maximum compression always corresponds to an average relative material density of approximately 0.55.

In this context, it should be mentioned that it was shown in Ref27that an inhomogeneous distribution of the initial density has a small influence on the field variables.

Friction between the tooling and the powder is another very important parameter at uniaxial die compaction. It effec- tively determines, among other things, the relation between the forces on the upper and the lower punches. In Ref28, a compre- hensive effort of measuring friction at compaction of hard metal

powders is presented, and those results are, as mentioned earlier, relied upon here.

SENSITIVITY ANALYSIS

The sensitivity measure used in this investigation is the so- called total effect defined by Sobol [35], and the sensitivity of the variables with respect to the objective function is shown in

Fig.4. In LS-OPT [34], this total effect is described by using sen- sitivity indices. The value of this index lies between zero and one for each of the variables and is determined with respect to the objective function. The sum of all the sensitivity indices is equal to one. Accordingly, if one variable has a sensitivity index with respect to the objective function equal to one, then the objective function is solely dependent on this variable and not on the others. In the pillars in^Fig.4(the longer the pillar, the more sensitive the variable), where the result from different sen- sitivity analyses are presented, the effect from each variable on each of the sub-objective functions is also indicated.

It is seen in^Fig.4that four variables are significantly more sensitive to the value of the target function than the rest, and, based on this, it is judged that these four variables are probably possible to determine with some accuracy using this approach.

The parameters chosen to be included in the optimization are shown in^Table3(and the parameters not included are shown in^Table4).

From the sensitivity analyses, it can also be concluded that the variables have different effects on the different sub-objective functions, indicating that the different sub-objective functions increase the capability to differentiate between the variables, thus increasing the possibility to determine them. In other words, if two variables show high correlation with respect to one of the sub-objectives, they can be less correlated with respect to another of these functions. This indicates that the experimental setup, with the instrumented die and its eight sen- sors, are useful compared to a setup measuring the average

FIG. 4 Sensitivity analysis including all constitutive parameters in the material model [19]. The variable sensitivity is presented based on sensitivity indices as defined in the text. The longer the pillar, the more sensitive the variable. The contribution from each sub-objective function (defined in the figure) is indicated within each pillar.

radial stress or strain, or to a setup measuring the radial stress only at one location.

VERIFICATION THROUGH INVERSE ANALYSIS USING SIMULATED DATA

The verification of the procedure was explained previously, and it was shown that if the method can effectively handle simulated data, it can also handle experimental data given that those are of comparable quality. Consequently, results from a situation where the “experimental data” is simulated and the solution (material parameter values) is known beforehand are presented and discussed below.

The conclusion from the sensitivity analysis shown previ- ously was that the parameters inTable3are the parameters in the constitutive model having a significant effect on the meas- ured output from the experiments, and these parameters are included in the inverse analysis with “simulated experimental data.” This is done from two different starting positions (differ- ent initial guesses on variable values) to see whether the method is robust in the sense that it converges to the same solution even if it is difficult to make an appropriate initial guess. Even though it is not explicitly shown in this text, the optimization method is indeed robust in this sense, as the same solution is found for different initial guesses.

The results are presented in the following way. The target forces and stresses, as described above, are compared to the

solution from the inverse analysis using simulated data as shown in^Fig.5. Clearly, the agreement is very good with small discrep- ancies in press force and radial stress distribution. In^Fig.6, the optimization histories of all the variables in the analysis of the simulated data are shown. The variables have converged to some degree, but the variable boundaries still allow for some small changes of the values. Obviously there is little room for improvement, but a slightly better solution is of course found if the number of iterations is increased further. The optimization history regarding variable values for the different iterations is also an indicator of the sensitivity, and thus confidence in the value found. From^Fig.6, it is possible to conclude that, first of all, the sensitivity analysis and the optimization histories both show that some of the variables are more sensitive than others.

Second, it is also shown that the most sensitive variable is the coefficientc₀₂in Eq11for determination of the yield stress in hydrostatic compression, indicating that the stress state in the powder is highly hydrostatic. Furthermore, it is also clear that the variableRis the least sensitive of these four variables. The variablec₀₁also shows a stable trend, which certainly gives some TABLE 3 Material parameters involved in the optimization

procedure.

Parameter Notation Unit

Intersection point of the failure and cap surfaces L/X

Aspect ratio of the elliptic cap R

Yield stress in hydrostatic pressure X(d) MPa

Note thatX(d) is described by two variables according to Eq11.

FIG. 5 Finite-element simulation of the die-compaction problem. Crosses correspond to results from a simulation with material characteristics according to Ref26. Full lines correspond to results from a simulation where the four most sensitive parameters (seeFig.4) have been determined from an inverse analysis as describe in the text. (a) Press force (upper curve) and friction force as function of the uniaxial compression of the powder. (b) Radial stress distribution,rr, at sensors 2, 5, and 8 as functions of the uniaxial compression of the powder. The upper curve corresponds to sensor 2.

TABLE 4 Material parameters used in the numerical calculations (but not involved in the optimization procedure).

Parameter Notation Value Unit

Initial average density q₀ 5000 kg/m³

Elastic Poisson’s ratio 0.37

Initial intensity of anisotropy e 1

Ratio on plastic shear strain ca 0.001

Ratio on plastic deviatoric work cg 4.5⁵

Ratio on plastic volumetric work cn 4.5⁵

Yield stress in uniaxial pressure Y(d) MPa

Elastic bulk modulus K(d) MPa

Back stress scale factor h

Flow angle offset / Degree

The parameters are discussed in the text and are pertinent to the Brandt and Nilsson [19] material model. The parameters related to the yield sur- face are defined in^Fig.2. Note thatci-parameters are used to define the intensity of plastic anisotropye.

confidence in the resulting value ofX(d). The variablesRandL/

X are also affecting the output but have a slightly less stable trend compared toc02. The results in^Fig.4, values in^Table5, and trends in^Fig.6all point in the same direction with regard to the variable that is the most sensitive and thereby the easiest one to determine, resulting in the most trustworthy value.

In ^Fig.7(a), the results for the material parameter X(d), determined by Eq11, are shown and compared to the ones used in the simulated data and the start values used in the optimiza- tion. The agreement is good for bothX(d) and the yield surface (as shown in^Fig.7(b)). It can be noted that the value ofX(d) is presented up to a relative density of 0.60. This limit comes from the fact that the experiments are performed up to an average relative density of approximately 0.55, but it has been noted in the simulations that, in some regions in the powder compact, the relative density reaches values of almost 0.60, and therefore

it is considered reasonable to show the values of the functions up to this value. The resulting variable values obtained from this analysis and from the known reference solution are pre- sented and compared in^Table5.

As can be seen inFigs.4and6,the measured output is sen- sitive to the hydrostatic stress at yielding, as well as to the parametersRandL/Xgoverning the flatness of the yield surface shown in^Fig.2. Also, the correlation betweenRandL/Xis con- siderable. This means that the values of these parameters can be changed simultaneously without changing the measured output much. The output is, however, not completely insensitive to such changes.

ANALYSIS OF MATERIALS A AND B

The results pertinent to a real experimental situation are pre- sented and analyzed by comparing the values identified for FIG. 6

Inverse analyses using simulated

“experimental” data as discussed in in the text. The optimization result at each iteration is shown as well as the boundaries of the variable space. (a) c01, (b) c02, (c)R, and (d)L/X.

TABLE 5 Resulting values on the optimized variables (from analysis using simulated data).

Parameter Notation Reference Value Solution

Intersection point of the failure and cap surfaces L/X 0.853 0.835

Aspect ratio of the elliptic cap R 0.928 1.045

Coefficient forX(d) c01 1248 1218

Coefficient forX(d) c02 0.134 0.135

Reference values are shown. These values are combined withkX¼0.6986 in Eq11to determineX(d).

material A with the ones for material B. In addition, the values determined for materials A and B can also be compared to the ones taken from Refs31and32, with the exception of the quan- titiesR andL/Xthat, in Ref26(and here), are set toR¼0.99 andL/X¼0.83 based on numerical findings and relevant values from the literature.

The initial values on the variables in the optimization pro- cedure, pertinent to material A, are the same as the ones listed in Refs31and32(with the exception ofRandL/Xas discussed above). These values are assumed to be physically reasonable for cemented carbides and not very far from the values to be determined for material A. Regarding material B, the initial val- ues are set to the final values found for material A.

The outcome of the analysis indicates that materials A and B are similar, but not identical. The results in^Fig.8(a)and^8(c) show that the simulated forces from the inverse analysis (and FEM) agree reasonably well with the experimental values for the respective materials. The discrepancy in friction force between the experimental results and the corresponding result from the inverse analysis is quite large initially but the situation improves at higher densities (which of course is preferable).

This is particular for material B where the difference between the two sets of force values is consistently less than 2 % at

higher densities, whereas the error is approximately one order higher for material A. Concerning the radial stress distribution, shown in ^Fig.8(b) and ^8(d), the agreement is good for both materials (in all cases except for sensor 8, the deviation between the two sets of results is consistently smaller than 5 %). This is true, both regarding the spread in radial stress along the press direction and the actual values across the density range studied.

Constitutively, the inverse analyses of materials A and B give solutions presented in ^{Table 6} (variable values) and in

Fig.9(material parameters and yield surface). In^Fig.9, selected resulting material characteristics for both materials A and B are explicitly shown as well as corresponding results for cemented carbides presented in Ref31. Obviously, the three sets of results show differences as expected but as there are no previously pre- sented results concerning constitutive characterization of mate- rials A and B, it is impossible to determine the accuracy of the present approach based on these curves. However, from the pre- vious discussion and the fact that the numerical procedure lead- ing to the results in ^{Table 6} and in ^{Fig. 9} could be carried through without any problems, some confidence is definitely gained with regard to the usefulness of the present approach for constitutive characterization of powder materials. In this con- text, some interesting results by Zhang et al. [39] should be mentioned. In Ref 39, characterization of a powder material using an instrumented cubic die is attempted showing promis- ing results. It would indeed be interesting to apply the present approach using such an experimental device, but this is, how- ever, left for future studies and presently only a cylindrical die is relied upon.

SKEWED UPPER PUNCH

In Ref27, a numerical analysis was performed to determine the influence from the geometry of the upper punch on the stress distribution in the powder compact at compaction. It was found that using a skewed upper punch resulted in a slightly larger radial stress variation (compared to the flat punch case), indi- cating that the skewed punch should be preferable for the pres- ent purposes. Basically, a skewed punch would introduce a more inhomogeneous stress distribution and thereby activate more constitutive parameters in the model. It should be empha- sized that this effect reported in Ref27was small, but, neverthe- less, noticeable. Accordingly, a sensitivity analysis in the same manner as above was performed but now with a skewed upper punch. The skewed punch is a plane rotated 20with respect to the plane perpendicular to the press direction. In short, the FEM results from this limited study indicate that introducing a skewed punch does not activate additional parameters (as compared with the flat punch case) in a noticeable manner.

UNLOADING

The unloading part of the press cycle was not analyzed in any detail in the present study. The reason for this is that the FIG. 7 Results from the inverse analyses using simulated “experimental”

data as discussed in in the text. The derived solution is compared to the reference and start state. Explicit values on the material parameters are presented inTable5, for both the reference solution and the solution to the inverse problem. (a) X(d), and (b) yield surface.

experimental efforts in Ref26were mainly directed toward the loading stage and unloading was not given much attention.

However, for completeness, a sensitivity analysis was performed to investigate whether the unloading stage could give any further information about relevant constitutive parameters.

Without going into detail, this study suggested that elastic parameters, bulk modulus, and Poisson’s ratio possibly could be determined by studying the unloading stage of the press cycle.

This matter is, however, left for future studies.

ANISOTROPY

It should be clearly stated once again that, in the present study, the constitutive description is taken from Refs31and32, which is a somewhat simplified version of the original model presented by Brandt and Nilsson [19]. Most importantly, in Ref19, there

were additional parameters influencing, as compared to Eq 13, the back stress scaling functionh(in this study,hdepends only on plastic anisotropic intensitye). From the present numerical results pertinent to simulations of die compaction, the anisot- ropy does not develop noticeably and the back stress therefore does not affect the solution as h takes on very small values.

However, as this feature is a result of a simplification, it can def- initely be of interest to determine whether the die-compaction solution would be significantly affected if anisotropy, and thereby the parameterh, is developing strongly during the test.

Consequently, a simulation, wherehis set to a very high value, was performed and it was found that the solution is in fact then strongly affected by such kinematic hardening effects. This is not something that will be analyzed further in the present con- text, but here it is simply stated that the current constitutive FIG. 8

Experimental data for materials A and B compared with finite-element results pertinent to the die-compaction problem. In the finite-element simulations, material parameters determined by the inverse analysis, discussed in the text, are used.

Crosses are results from experiments presented in Ref26. Full lines correspond to finite-element results with material characteristics determined from the inverse analysis. Results for material A are shown in a and b, and corresponding ones for material B in c and d. (a) Press force (upper curve) and friction force as functions of the uniaxial compression of the powder. (b) Radial stress distribution,rr, at sensors 2, 4, 6, and 8 as functions of the uniaxial compression of the powder. The upper curve corresponds to sensor 2. (c) Press force (upper curve) and friction force as functions of the uniaxial compression of the powder. (d) Radial stress distribution,rr, at sensors 2, 4, 6, and 8 as functions of the uniaxial compression of the powder. The upper curve corresponds to sensor 2.

TABLE 6 Resulting values on the optimized variables (from analysis using real experimental data from Refs26,31, and32).

Parameter Notation Material A Material B Refs26,31, and32

Intersection point of the failure and cap surfaces L/X 0.82 0.80 0.833

Aspect ratio of the elliptic cap R 1.14 1.16 0.990

Coefficient forX(d) c01 1061 1086 1511

Coefficient forX(d) c02 0.141 0.138 0.136

These values are combined withkX¼0.6986 in Eq11to determineX(d).

description may not accurately describe the effect of the kine- matic hardening, not even in a fairly simple die-compaction situation.

Concluding Remarks

In the present investigation, material characterization of cemented carbide powder materials has been attempted based on uniaxial die-compaction experiments in combination with inverse modeling. The radial stress distribution at the die wall is determined at a discrete number of points during the experi- ments and subsequently used in the inverse analysis together with experimentally determined press forces. The most impor- tant conclusions from this study can be summarized as follows:

• The experimental setup produces results of sufficient accuracy to be used for inverse modeling.

• The present approach is shown to be a useful tool for reducing the number of experiments required for a com- plete characterization of WC/Co powder material.

• Changing the shape of the punches in the experimental setup does not in a noticeable manner contribute to addi- tional information about the constitutive parameters.

• The numerical results indicate that additional constitutive parameters can be determined by also applying the pres- ent inverse methodology to the unloading stage of the press cycle. The details about this are left for future studies.

• Additional experiments must be performed for a com- plete characterization of the WC/Co powder materials.

Finally, it should be emphasized that constitutive character- ization of powder materials using inverse methods has been attempted previously in the literature. The main novelty about the present approach, though, concerns the fact that by meas- uring the radial stress distribution at the die wall continuously during the experiment, it is also possible to apply these methods in the case of a very complicated constitutive description with many material parameters and functions.

ACKNOWLEDGMENTS

This work was performed within the VINN Excellence Center Hero-m, financed by VINNOVA, the Swedish Government Agency of Innovation Systems, Swedish Industry, and KTH (the Royal Institute of Technology). The writers thank Bo Jansson, Stefan G. Larsson, and Dirk Sterkenburg, Seco Tools AB, and Hjalmar Staf, Sandvik Coromant AB, for many interesting dis- cussions and also for providing the materials.

APPENDIX

The sensitivity analysis, including the finite-element simula- tions, is controlled using LS-OPT [34]. LS-OPT [34] is a simula- tion module designed for analysis of the design space of a problem, including optimization, trade-off analysis, and sensi- tivity analysis. It is then coupled with an arbitrary solver, in this case a finite-element solver (LS-DYNA [33]), which is directed by LS-OPT [34] to run simulations with specific parameter val- ues. LS-OPT [34] then uses the results from the FE-simulations for, as in the present case, a sensitivity analysis. LS-OPT [34]

and LS-DYNA are developed by the same company, Livermore Software Technology Corporation (LSTC), and has therefore a prepared interface between them. This means in practice that LS-OPT [34] can in many cases directly read and handle the resulting files produced by a simulation using LS-DYNA [33], without the need for extensive programming. However, LS- OPT [34] can be used together with any other solver. LS-OPT [34] detects signals from LS-DYNA [33] concerning the status of the simulations, and invokes the post-processing commands and later the sensitivity analysis commands or optimization commands when the simulations are completed.

To execute the compaction simulation using LS-DYNA [33] and a user subroutine material model, it is necessary to compile the user subroutine into the executable LS-DYNA [33]

file before starting the simulation. The main command that is FIG. 9 Results from the inverse analyses using real experimental data as

discussed in the text. The derived solutions for material A and B are compared to the findings by Brandt [31,32] (the values on the parametersRandL/Xare not according to Refs31and32, but instead have values R¼0.99 andL/X¼0.83 as discussed in Ref26).

Explicit values on the appropriate material parameters are presented inTable6. (a) X(d), and (b) yield surface.

used in these simulations to control the adaptive meshing is

*CONTROL_ALE. It is used to keep the integrity of the finite elements intact by keeping their size relatively equal to the neighboring elements. The option controlling this type of ALE variant, in this command, is called BFAC, and is here set to 1.

The reason for this is that other settings tested, in the CONTROL_ALE-command, results in fewer elements toward the contact interface (between the material and the die) where some of the most important results are determined.

For the sensitivity analysis and the optimization, LS-OPT [34] is equipped with a graphical user interface (GUI). In this GUI, it is possible to control almost all of the features incorpo- rated into the program, sometimes just by checking boxes. It is also possible to state commands in a text file. It is then neces- sary to state the type of surrogate model to be used (solver order RBF) and the way in which the variable values to be simulated are chosen (solver experiment design space_filling).

Also, it is necessary to specify the solver executable, the input file, and the possible flags needed. It is sometimes neces- sary to specify a pre-processor, active prior to the simulation presenting the solver with the solver-specific input data format- ted for the particular solver. Accordingly, commands used to specify the solver (solver command “path to solver executable”) and its input file (solver input file “path to solver input file”) must be included in the LS-OPT [34] input file. If a pre- processing step is needed before the simulations, then this pre- processor needs to be specified with regard to the executable (prepro command “path to pre-processor”) type (prepro own) as well as its input file (prepro input file “path to pre-processor input file”). To call the search algorithm consisting of first a genetic algorithm and then a gradient method, it is necessary to issue the command (optimization algorithm hybrid ga - Use GSA) in the LS-OPT [34] input file.

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