## Abstract

Aluminum powder compaction was studied using both test and simulation. Cold compaction, hot compaction, and vibration-assisted (cold) compaction tests were conducted to achieve different density ratios. First, the hot compaction test (at 300 °C, compression pressure 140 MPa) improved about 6% compared with cold compaction under the same compression pressure. Second, although the relative density ratio does not obviously improve at a vibration-assisted (cold) compaction, the strength of the specimens made under vibration loading is much better than those of cold compaction. Additionally, finite element models with well-calibrated Drucker–Prager Cap (DPC) material constitutive model were built in abaqus/standard to simulate the powder compaction process. The results of the finite element model have very good correlations with test results up to the tested range, and this finite element model further predicts the loading conditions needed to achieve the higher density ratios. Two exponential equations of the predicted density ratio were obtained by combining the test data and the simulation results. A new analytical solution was developed to predict the axial pressure versus the density ratio for the powder compaction according to DPC material model. The results between the analytical solution and the simulation model have a very good match.

## 1 Introduction

Powder compaction technique attracted many researchers’ notice when they were developing the new material because of its advantage, as stated by Thummler and Oberacker [1]. For example, Ghorbani et al. [2] fabricated Cu-Cr, Cu-Cr/CNT powders and compressed them to the samples under different temperature conditions; it was found that the elevated temperature had a significant effect for the improvement of the hardness. Rahmani et al. [3] used a dynamic compaction method with a modified Hopkinson Pressure bar to achieve the samples with a relative density of more than 99%. The results showed that the micro-hardness and compressive strength of the samples increased by about 70% and 50%. Saboori et al. [4] produced the Al matrix with graphene nanoplatelets by the conventional powder metallurgy. The fabricated nanocomposites showed a high hardness by increasing the compression pressure. Wu and Kim [5] fabricated aluminum alloy 6061 with carbon nanotube composites through the semi-solid powder processing technology. They investigated the effects of the processing temperature for the fracture surface and the hardness of the composites. And the aluminum alloy 6061/CNT composites showed a good densification above 99%. Yim et al. [6] used the static and shock wave compaction technique to compress CoCrFeMnNi high-entropy alloy, the relative density of the sample made by the shock wave compaction reached to 95%, after sintering, to 99.5%. Shokrollahi and Janghorban [7] summarized a review of the property, characteristics, and applications about soft-magnetic composite materials. And they found that soft-magnetic composites were generally manufactured by the powder compaction combined with the warm compaction and two-step compaction. Liu et al. [8] prepared the reduced graphene oxide (rGO) and the graphene nanosheets (GNSs) reinforcement of aluminum matrix nanocomposites (AMCs) through the coating. And they made use of powder compaction method to manufacture disc-shaped specimens and then sintered in the inert atmosphere. The hardness was obviously improved by 32% and 43%, respectively, for the 0.2 wt% rGO-Al and 0.15 wt% GNSs-Al composites.

Many researchers investigated the powder compaction by combining experiments and simulation models. Wu et al. [9] used finite element methods (FEM) and Drucker–Prager Cap (DPC) model to investigate the variation of the pharmaceutical powder’s relative density with the applied pressure. And the uniaxial compaction experiments were used to calibrate the DPC model. A good agreement between the experimental and the FEA results was achieved, which indicated that the FEM model can predict the major features of the powder mechanics behavior during the compaction process. Khoei et al. [10] proposed a double-surface plasticity theory to simulate the powder compaction process. The numerical simulations were conducted to well predict the density of the material. Shang et al. [11] presented a calibration procedure for the powder compaction simulation. A new die with the radial sensors was designed to help the calibration of the material parameters. Heckel [12] conducted some powder compaction tests for the metallic materials to obtain the density with the compression pressure curves under 103–206 MPa. The test results provided a relation between the density of the material and the punch pressure. Sinka et al. [13] developed a high-pressure tri-axial testing facility to study the characteristics of the powder compaction, where the density of the specimens was very close to the full density of the material.

Recently, spark plasma is one of the most advanced powder compaction methods, which can significantly improve the material density at a very fast speed under the high-temperature condition during the powder compaction process. Guillon et al. [14] reviewed the field-assisted sintering technology and described the working principles and the historical background for it. And they discussed the refractory materials and the nanocrystalline ceramics and introduced the concepts of the advanced tool. However, the cost of the spark plasma instrument is very high compared to that of others. Rothe et al. [15] proposed a compressible thermo-viscoplastic constitutive model based on the multiplicative decomposition of the deformation gradient for the field-assisted sintering technology. They discussed the required experiments that can help obtain the electrical, thermal, and mechanical properties for the materials. They identified the material parameters for this constitutive model. Dash et al. [16] compared the maximum Vickers hardness under the conventional and the spark plasma technique for Cu-Al_{2}O_{3} metal matrix composites. The maximum hardness is 60, 70, and 80 for Cu-15 vol% Al_{2}O_{3} sintered conventionally in N_{2}, Ar, and H_{2} atmosphere, respectively. And the hardness reached 125 for Cu-5 vol% Al_{2}O_{3} nanocomposite made by the spark plasma sintering technique. The conclusion shows that the spark plasma sintering technique has a very good advantage when compared with the conventional sintering techniques.

For the numerical simulation study of the powder compaction, the discrete element method (DEM) and finite element method (FEM) are used to predict the density of the material and study its mechanics mechanism. DEM builds the real material particles to simulate a dynamic process of the powder compaction, which is based on the Coulomb friction theory. Garner et al. [17] used the DEM model to investigate the micromechanical characteristics of the powder compaction process. A new adhesive elastoplastic contact model was proposed to define the force–displacement characteristics between the contacting particles under the high confining conditions. FEM models based on continuum mechanics also can accurately predict the density of the material. Two widely used constitutive models in FEM are Drucker–Prager model and Mohr–Coulomb model. Cante et al. [18] used the elastic–plastic Drucker–Prager model to predict the behavior of the powder compaction and proposed a set of calibration experiments for the powder compaction simulation model. Bier et al. [19] used an analytical solution and some instrumented tests to study the relative density of the powder compaction. Most of the above-mentioned models need extensive experimental data to calibrate the material parameters in the constitutive models. Mamalis et al. [20] discussed and introduced some theoretical and empirical models under the dynamic powder compaction in order to understand the mechanics mechanism of the powder compaction. Parilak et al. [21] developed a practical compaction equation for the cold powder compaction through running 205 various metal powder mixes, which can give a direct calculation of the compaction force for various powders. Feng et al. [22] made use of multi-particle finite element method (MPFEM) and cohesive zone method (CZM) to simulate the compaction process of Al and NaCl laminar composite powders. The fracture phenomenon of the brittle powder materials was modeled by CZM with the initial cracks. They found that the CZM-based MPFEM could find the internal mechanics mechanism of the powder compaction process. It includes the particle rearrangement, the plastic deformation of Al particles, and the brittle fracture of NaCl particles.

Recently, a vibration-assisted compaction method was proposed as a potential way to improve the material density of the specimen made from the powder compaction under the room temperature condition. This method was widely used to improve the material density for the particle industry in the 1960s. An et al. [23] designed an instrument to investigate the cold compaction of the cooper powders with the vibration loading. And they concluded that vibration loading could improve the density and uniformity of the material. Fartashvand et al. [24] used an ultrasonic vibration method to enhance the density of titanium. It was found that a fine size powder can give an ideal density under the ultrasonic vibration loading. The friction force between the die wall and the powder was reduced through the vibration loading. Majzoobi et al. [25] manufactured Al7075-SiC nanocomposites by the hot dynamic compaction. They obtained a similar conclusion that the relative density of the SiC reinforced with Al7075 was slightly reduced by 2%. However, the compressive strength of the material was increased by about 60%.

There are some unique features in our work presented in this paper. For hot compaction tests, we developed an alternative heater device to provide an elevated temperature, which is more affordable than an environment chamber and the spark plasma instruments. For the vibration-assisted powder compaction tests, most researchers have to develop the special instruments to run the dynamic tests. Instead, we customized our MTS universal testing machine so that it can provide the vibration loading with different force magnitudes, different frequencies, and different dwell times. Although the finite element model can be used to simulate the powder compaction process, researchers have to calibrate their models with many tests. A different calibration method was brought up to increase the efficiency of obtaining the material parameters. Prediction of the compression pressure for a specific relative density ratio of the specimen is important before running the powder compaction tests. There are only a few works that combined the extensive tests and the simulations for the prediction of the density ratios, which will be presented as a key part of this paper.

The purpose of this paper is to find a better way to improve the relative density of specimens made from powder compaction in a lab scale and to validate that the DPC model can simulate the aluminum powder compaction process. From our investigations, a good match between test data and simulation model results was achieved for the cold and hot compaction processes (300 °C). The vibration-assisted compaction method and the hot compaction method were used to enhance the relative density of aluminum specimens compared to the cold one. Both of them have a significant improvement in material strength or the relative density. Finite element models were built to simulate the powder compaction processes in order to study their mechanics mechanism. Finally, a new analytical solution was developed to predict the axial pressure versus density ratio for powder compaction. This will provide a good guideline for researchers to run powder compaction tests with desired density ratios.

## 2 Powder Compaction Tests

### 2.1 Design of Die and Test SetUp.

Die assembly is composed of a base, a die, and a punch. Figures 2(a) and 2(b) show the die assembly and the test setup of the powder compaction. The raw material of the die is made from AISI 1144 carbon steel rod which is easy to machine. This material has a strong hardness and the maximum yield strength 689 MPa.

For the cold compaction tests, a hydraulic MTS system with a maximum force of 110 kN was used to provide the compression pressure for the aluminum powder compaction test. The die assembly was put on the downside platen of the MTS machine (see Fig. 2(c)). After the aluminum powder was put into the hole of the die. The punch was loaded by the MTS machine to finish the powder compaction test.

For the hot compaction tests, an extra induction heater was put on around the die. Figure 2(d) shows the setup of the hot compaction test. The heater device is composed of a heater, a PID controller, and a thermal couple. The heater model is HBA-202040. The diameter and the height of the heater are 50 mm × 50 mm. It is just suitable for the size of the powder compaction die. Thus, heat energy is easily transferred to the die. Because there is only a little space between the heater and the die. PID controller (CN470) is used to control the temperature of the die. The thermal couple is a type K washer thermal couple.

### 2.2 Experiment Methods.

Three experiment methods were studied in this paper. They are cold compaction, hot compaction, and vibration-assisted compaction. It should be noted that oxidation was not obviously observed in our material samples although shielding gases have not used yet, which is planned for the next stage while fabricating large material samples. SEM images and the mechanical strength tests for the compacted samples are also provided for the analysis.

#### 2.2.1 Cold Compaction Test.

The force control mode of the MTS system was used to apply the compression pressure onto the powder. It is helpful to protect the universal test system away from the damage.

A maximum force of 10 kN was applied to the aluminum powder for the cold compaction test. The powder mass is 0.427 g. After the maximum force was reached, the compression pressure was kept for 300 s. This hold action is useful to make the quality of the specimen better. The initial and final height of powder geometry are 0.632 cm and 0.266 cm, respectively.

#### 2.2.2 Hot Compaction Test.

For the hot compaction test, the aluminum powder mass is 0.445 g, and the compression force is also 10 kN. The temperature of the die was steadily increased by the heater. After 2 h, it reached to 300 °C. The maximum compression pressure is 140 MPa. The hold time is also 300 s. After unloading the MTS system and turn off the heater, the die and the specimen cooled down. One interesting thing is that the specimen was not easily separated from the compression punch because of the surfaces glued between the specimen and the punch. This phenomenon shows that the elevated temperature is obviously useful to make the atoms move dramatically. Thus, the interfaces bonded perfectly. The initial and final heights of hot powder are 0.590 cm and 0.258 cm, respectively. The cold and hot compaction test data are summarized in Table 2.

Cold powder compaction (room temperature) | Hot powder compaction (300 °C) | |
---|---|---|

Force (kN) | 10 | 10 |

Mass (g) | 0.427 | 0.445 |

Height (cm) | 0.266 | 0.258 |

Diameter (cm) | 0.96 | 0.96 |

Volume (cm^{3}) | 0.192 | 0.186 |

Density (g/cm^{3}) | 2.22 | 2.39 |

Initial relative density ratio | 0.35 | 0.38 |

Compaction pressure (MPa) | 140 | 140 |

Relative density ratio | 0.82 | 0.88 |

Cold powder compaction (room temperature) | Hot powder compaction (300 °C) | |
---|---|---|

Force (kN) | 10 | 10 |

Mass (g) | 0.427 | 0.445 |

Height (cm) | 0.266 | 0.258 |

Diameter (cm) | 0.96 | 0.96 |

Volume (cm^{3}) | 0.192 | 0.186 |

Density (g/cm^{3}) | 2.22 | 2.39 |

Initial relative density ratio | 0.35 | 0.38 |

Compaction pressure (MPa) | 140 | 140 |

Relative density ratio | 0.82 | 0.88 |

From Table 2, one can conclude that the density ratio of the specimen made from the hot powder compaction is larger than that made from the cold powder compaction (from 0.82 to 0.88) under the same compaction pressure. Thus, the elevated temperature compaction test is one effective way to enhance the density ratio for the aluminum powder compaction.

#### 2.2.3 Vibration-Assisted Compaction Test.

For the vibration-assisted compaction test, the force control mode in the MTS testing machine was used. The loading histories of the quasi-static loading (or called monotonic loading) and the vibration-assisted one are shown in Fig. 3. The individual and the combined force–displacement curves are plotted.

Different loading conditions were programmed with the MTS system, using the force control mode. For the four groups of quasi-static and vibrations tests, the following factors were considered, force amplitude, powder mass, quasi-static loading pressure, vibration loading pressure, and dwell time of the powder compactions. Each group of tests contains two or three tests of vibration and/or quasi-static loading. Although the relative density ratio did not change very much for the vibration loading condition when compared with the quasi-static loading condition, the strength of the specimen compacted from the vibration loading condition is higher than that made from the quasi-static loading condition. This part will be described in detail in Secs. 2.2.4 and 2.2.5. The following paragraphs introduce these four groups of tests and a summary of the loading test data (see Table 3).

Test number | Force and loading condition | Mass (g) | Density (g/cm^{3}) | Relative density (–) | Static pressure (MPa) | Vibration pressure (MPa) | Dwell time (s) |
---|---|---|---|---|---|---|---|

1 | 10 kN + 300 °C | 0.445 | 2.39 | 0.88 | 140 | 0 | 300 |

2 | 10 kN | 0.427 | 2.22 | 0.82 | 140 | 0 | 300 |

3 | 9 kN + 1 kN + 10 Hz | 0.543 | 2.20 | 0.82 | 124 | 13.8 | 300 |

4 | 14 kN | 0.530 | 2.29 | 0.85 | 197 | 0 | 300 |

5 | 9 kN + 5 kN + 10 Hz | 0.360 | 2.35 | 0.87 | 128 | 72 | 300 |

6 | 9 kN + 5 kN + 20 Hz | 0.329 | 2.39 | 0.89 | 137 | 95 | 300 |

7 | 16.5 kN | 0.257 | 2.37 | 0.88 | 233 | 0 | 1800 |

8 | 9 kN + 7.5 kN + 20 Hz | 0.264 | 2.34 | 0.87 | 127 | 106 | 1800 |

9 | 10 kN | 0.973 | 2.01 | 0.74 | 141 | 0 | 300 |

10 | 9.8 kN + 0.2 kN | 0.965 | 2.05 | 0.76 | 138 | 3 | 300 |

Test number | Force and loading condition | Mass (g) | Density (g/cm^{3}) | Relative density (–) | Static pressure (MPa) | Vibration pressure (MPa) | Dwell time (s) |
---|---|---|---|---|---|---|---|

1 | 10 kN + 300 °C | 0.445 | 2.39 | 0.88 | 140 | 0 | 300 |

2 | 10 kN | 0.427 | 2.22 | 0.82 | 140 | 0 | 300 |

3 | 9 kN + 1 kN + 10 Hz | 0.543 | 2.20 | 0.82 | 124 | 13.8 | 300 |

4 | 14 kN | 0.530 | 2.29 | 0.85 | 197 | 0 | 300 |

5 | 9 kN + 5 kN + 10 Hz | 0.360 | 2.35 | 0.87 | 128 | 72 | 300 |

6 | 9 kN + 5 kN + 20 Hz | 0.329 | 2.39 | 0.89 | 137 | 95 | 300 |

7 | 16.5 kN | 0.257 | 2.37 | 0.88 | 233 | 0 | 1800 |

8 | 9 kN + 7.5 kN + 20 Hz | 0.264 | 2.34 | 0.87 | 127 | 106 | 1800 |

9 | 10 kN | 0.973 | 2.01 | 0.74 | 141 | 0 | 300 |

10 | 9.8 kN + 0.2 kN | 0.965 | 2.05 | 0.76 | 138 | 3 | 300 |

The first group of tests is the 10 kN quasi-static loading and the 9 kN quasi-static force with 1 kN vibration force. For the vibration loading condition, the vibration frequency is 5 Hz, and the dwell time is all 300 s. The mass is 0.427 g for the quasi-static loading, and the mass is 0.543 g for the vibration loading. The relative density is 0.82 for the quasi-static test, and the relative density is 0.82 for the vibration test. It seems that they have almost the same density ratio. The vibration effect on the powder compaction is not obvious. The reason may be that the current vibration amplitude and frequency are not enough to improve the density of the powder compaction. Thus, for the second group of tests, the vibration frequency and amplitude were enhanced to 5 kN and 20 Hz. This means that more energy from the vibration loading was applied to the powder.

For the second group of tests, three tests were done. They are the quasi-static loading 14 kN, the quasi-static loading 9 kN with the 5 kN vibration loading with 10 Hz frequency, and the quasi-static loading 9 kN with the 5 kN vibration loading with 20 Hz frequency. The dwell time is all 300 s. The quasi-static loading 14 kN represented 128 MPa pressure. And the vibration force represented 72 MPa pressure. The final mass of the powder is 0.53 g, 0.36 g, and 0.329 g, respectively. The relative densities of the specimens are 0.87, 0.89, and 0.91. It seems that the powder with the vibration loading with 20 Hz has a better relative density. But the real force amplitudes for the two vibration loadings are bigger than the input value (14 kN). Actually, they have the same input total force value 14 kN (9 kN+5 kN). But the MTS system generated a different bigger force value 16.4 kN instead. The relative density for the quasi-static force 9 kN with the vibration frequency 10 Hz is higher than that of the quasi-static loading obviously.

For the third group of tests, the quasi-static force is 16.5 kN, and the vibration force is the quasi-static force 9 kN and the 7.5 kN vibration force with the vibration frequency 20 Hz. It means a 127 MPa quasi-static pressure and a 106 MPa vibration pressure. The final masses of the powder are 0.257 g and 0.264 g, respectively. The dwell time is 1800 s in this test in order to verify the time effect for the relative density. Eventually, the relative density didn’t improve.

For the fourth group of tests, the quasi-static loading is 10 kN, and the vibration loading is 9.8 kN quasi-static loading with 0.2 kN vibration force with a vibration frequency of 10 Hz. The quasi-static compression pressure is 130 MPa, and the vibration compression pressure is 3 MPa. The powder masses are 0.973 g and 0.965 g, respectively. The purpose of this group of tests is to investigate the effect of the powder mass. The two tests have almost the same mass. The dwell time is also 300 s. The relative density of the specimen with the vibration loading is a little bit higher than of the specimen without the vibration loading. They are 0.76 and 0.77, respectively.

Table 3 gives a summary of the aluminum powder compaction tests from both the quasi-static loading and the quasi-static force with the vibration force. The hot compaction test data is also included. Figure 4 shows the relative density ratio of the aluminum powder under different loading conditions. From Table 3 and Fig. 4, the relative density ratios did not improve dramatically under the vibration loadings in the current aluminum powder compaction tests. This does not mean that the vibration loading can not enhance the density of the material. In fact, a higher frequency for the vibration loading may be helpful for the improvement of the density. The capacity of the vibration frequency in our MTS machine is very limited, which is one of the reasons that the relative density ratio was not improved. The dwell time may be also another reason.

#### 2.2.4 Scanning Electron Microscopy Pictures of Compacted Samples.

For the loading 14 kN quasi-static force and the loading 9 kN quasi-static force with 5 kN vibration loading with 10 Hz frequency, SEM images are shown in Fig. 5. SEM images can provide micro-structural information of the specimen. The dark area is the pore in the section of the specimen. The darker areas the specimen had, the smaller the density ratio of the specimen became. From Fig. 5, the dark area of the specimen section in the quasi-static loading is more than that of the specimen section in the vibration loading. Thus, the relative density ratio under the quasi-static loading is smaller than the one under the vibration loading at the observed spot. Image analysis shows that the ratio of the dark area to the white area under the quasi-static loading is 6.2% and that under the vibration loading is 5.5%. From SEM pictures, the difference in the density between the quasi-static and the vibration loading has the same conclusions as the direct measurement (see Table 3). Another observation is that the void/pore description is more uniform in samples made from vibration compaction.

#### 2.2.5 Mechanical Strength Tests of Compacted Specimens.

Two groups of compression tests were conducted to compare the strength of specimens. The specimens for Group 1 were compacted with the maximum force of 16.5 kN. The test setup is shown in Fig. 6, which is the radial compression. One specimen was made from the vibration loading, and the other specimen was made from only the quasi-static loading. Figure 6 shows the force versus displacement curves of the two specimens. It is found that the specimen made from the vibration compaction has a better strength than that made from the quasi-static loading. However, their density ratios are almost the same. Note that the initial stiffness may not be accurately measured due to uncertainties from initial surface contact of radial compression tests.

The specimens for Group 2 are compacted with the maximum force 10 kN, namely the quasi-static loading force 10 kN and the quasi-static loading force 9.8 kN with the vibration loading 0.2 kN force. Figure 7 shows the specimen deformation after the compression test in the MTS machine. This setup picture is different from Fig. 6. Figure 6 is a radial compression test. Figure 7 is an axial compression test. Both the strength comparison of the specimens and the strain-stress curves under the axial compression loading are illustrated in Fig. 7.

It is found in Fig. 7 that the strength of the specimen compacted under the vibration loading is higher than that of quasi-static loading. The maximum force magnitude is increased from 4000 N to 4500 N for the specimen made from the vibration compaction. Accordingly, the maximum stress magnitude is increased from 55 MPa to 65 MPa.

## 3 Finite Element Simulation of Aluminum Powder Compaction

A finite element model was built to simulate the aluminum powder compaction process for the cold compaction and the hot compaction with 300 °C. This model will help to understand the mechanics mechanism of the aluminum powder compaction. The force with displacement curves can be predicted through this FE model. The relative density of the aluminum specimen also can be obtained from the simulation model. It will give a guide for the research of the powder compaction tests. For example, if a specific force loading is applied to the powder, this model can predict the density of the specimen made from the powder without running tests.

This finite element model was built by abaqus/standard (version 2017). The compaction die is a cylindrical body, shown in Fig. 2. The finite element model of the powder compaction can be built with an axisymmetric model. It was composed of a die and the aluminum powder. The die was considered as a rigid body, and the aluminum powder was considered as a deformable body. The CAX4R element is used in abaqus for the aluminum powder, which is a four-node axisymmetric quadrilateral, reduced integration, hourglass control element. The discrete rigid element RAX2 is used for the rigid die.

### 3.1 Model Assumptions and Boundary Conditions.

In the finite element model, the punch is considered as a rigid plane. The top surface of the powder body is loaded by an initial displacement. The initial displacement loading can also be replaced by the initial force loading. The results of the finite element model will be the same. Figure 8 shows the finite element model setup.

The die is also considered as a rigid body, and the powder is a deformable body. A deformable die model with steel properties was built to validate the accuracy of this assumption. From FE simulation, the die deformation is really negligible, so the rigid die is a valid assumption.

Figure 2 shows the geometry models of the cold compaction and the hot compaction tests. The finite element model setup is shown in Fig. 8, and the boundary conditions are shown in Fig. 9. The mesh size is 0.1 mm for the aluminum powder and the die. There is a contact definition between the die and the aluminum powder. The aluminum powder gradually slides along the die when subjected to a compaction pressure. A friction coefficient of 0.1 is defined in this model. The aluminum powder is applied by a prescribed displacement from the test data.

### 3.2 Constitutive Model of Aluminum Powder.

DPC model is a popularly used material constitutive model for the powder compaction simulation. Aluminum powder is considered as the compressible substance since there are a lot of pores inside. Drucker Prager Cap model can well describe the compressible characteristics of the powder.

The yield locus of the DPC model in the *p–t* plane is shown in Fig. 10. Uniaxial compression and confined compression tests (using our powder compaction tests as an approximation) were conducted to calibrate this model.

*β*and

*d*are material’s friction angle and the cohesion, which usually depends on temperature. The deviatoric stress measure

*t*is written as.

*p*= −(1/3)

*trace*(

*σ*) is the mean pressure, $q=(3/2)S:S$ is the Mises equivalent stress,

*r*= ((9/2)

*S*:

*S*·

*S*)

^{1/3}is the third deviatoric stress invariant (

*s*=

*σ*+

*pI*), and

*K*is a material parameter that defines the dependence of the yield surface on

*r*.

*R*is a material parameter that can give the shape of the cap and

*p*

_{a}is an evolution parameter that describes the volumetric plastic strain. The hardening law is a linear function of hydrostatic compression yield stress (

*p*

_{b}) and volumetric inelastic strain, which is described as follows:

*p*

_{a}is described as follows:

The material parameters of Drucker–Prager model contain the material cohesion *d*, the material angle of friction *β*, the cap eccentricity *R*, the initial cap yield surface position on the volumetric inelastic strain axis $\epsilon volin\u22230$, the transition surface radius parameters *α*, and the ratio of the flow stress in triaxial tension to the flow stress in triaxial compression *K*. The above six parameters mentioned are Drucker–Prager/Cap plasticity yield surface parameters. Elastic modulus and Poisson ratio are also needed for this model. The key input is the evolution of the volumetric strain hardening curve (Eq. (4)).

### 3.3 Calibration Method of Drucker–Prager Cap Material Constitutive Model.

An iterative calibration method combining experiment and simulation was used to obtain the material parameters of the DPC model. First, to conduct a confined compression test to get DPC volumetric strain hardening parameter curve, then to perform finite element analysis to obtain the unknown material parameters of the DPC model, and finally, to iterate finite element analysis until the force–displacement curve of the simulation model gets well match with that of the test. Figure 11 shows a flowchart of this calibration method.

### 3.4 Material Property of Aluminum Powder.

According to the calibration method of Drucker–Prager Cap model (see Sec. 3.3), the material properties of the aluminum powder are listed in Table 4. In this paper, two FEA models are built, which are the loading condition 10 kN force on 0.427 g aluminum powder without temperature loading, and the loading condition 10 kN force on 0.445 g aluminum powder with 300 °C temperature loading. The same elastic material property and the plastic yield parameters are used as listed in Table 4. The difference between the cold compaction and the hot compaction models is that these two models have different volumetric strain hardening curves. Figure 12 shows the volumetric strain hardening curves for these two loading conditions.

Cap plasticity’s parameters | |

Cohesion d (MPa) | 0.01 |

Friction angle β (deg) | 71 |

Eccentricity R | 0.021 |

Initial yield surface position p_{a} (MPa) | 0.01 |

Flow stress ratio K | 1 |

Elastic parameters | |

Young's modulus E(MPa) | 5000 |

Poisson ratio μ | 0.29 |

Cap plasticity’s parameters | |

Cohesion d (MPa) | 0.01 |

Friction angle β (deg) | 71 |

Eccentricity R | 0.021 |

Initial yield surface position p_{a} (MPa) | 0.01 |

Flow stress ratio K | 1 |

Elastic parameters | |

Young's modulus E(MPa) | 5000 |

Poisson ratio μ | 0.29 |

### 3.5 Results Comparison.

For the results of the finite element model, the top surface of the aluminum powder model can obtain the total force of all nodes with time. And the displacement with the time curve is the input excitation in the simulation model. Thus, the force with the displacement curve can be obtained. Figure 13 shows the force with displacement curves between tests and finite element models for the cold compaction and the hot compaction. One can conclude that there is a good match between tests and simulations. In the initial phase of powder compaction, powders experienced elastic deformation because of some loose powders, and then the plastic strain gradually increases with the growth of the displacement. The evolutions of the relative density ratios with respect to the axial pressure are plotted in Fig. 14. It is found that it needs a larger force for the cold compaction test than that of the hot compaction in order to obtain the same density ratio. However, there is a transition point at the beginning of the curves (see Fig. 13), which may be due to the thermal expansion or the friction between the die and the punch.

The one important phenomenon is that the relative density ratio of the hot compaction test is larger than that of the cold compaction test. The biggest difference in the density ratio between the cold compaction and the hot compaction is about 0.15. So, the hot compaction test will be a good choice for the improvement of the relative density ratio of the material if the cold compaction test can not reach an ideal objective.

## 4 Analytical Solution of Axial Pressure Versus Density Ratio

*H*is the height of a powder sample. Note that the elastic strains are very smaller than plastic strains, so they are negligible for powder compaction analysis. From the definitions of volumetric plastic strain rate ($\epsilon \u02d9vol$) and equivalent plastic strain rate ($\epsilon \u02d9eq$), we can get $\epsilon \u02d9vol=\epsilon \u02d9zz$, and $\epsilon \u02d9eq=(2/3)\epsilon \u02d9ij\u2032\epsilon \u02d9ij\u2032=(2/3)\epsilon \u02d9zz$, where $\epsilon \u02d9ij\u2032$ are deviatoric plastic strain rate components. Hence, we have the following equation:

*γ*=

*R*/(1 +

*α*−

*α*/cos

*β*), and

*p*

_{b}−

*p*

_{a}=

*R*(

*d*+

*p*

_{a}tan

*β*). Since this an axial symmetric loading in powder compaction,

*K*= 1, then

*t*=

*q*. Therefore, Eq. (3) becomes

*p*/∂

*σ*

_{ij}) = (1/3)

*δ*

_{ij}and (∂

*q*/∂

*σ*

_{ij}) = (3/2

*q*)

*s*

_{ij}are used. From Eq. (10), the volumetric strain rate and equivalent plastic strain rates can be evaluated as follows:

From the feature of a volumetric strain hardening curve (see Fig. 12), it is assumed that $pb=A+Bec\epsilon vol$, where *A*, *B*, and *C* are three curve fitting constants, as listed in Table 5. Note that the best curve fitting gives the parameter *A* = 0, which is actually a very small initial number.

Cold compaction | Hot compaction | |
---|---|---|

A (MPa) | 0 | 0 |

B (MPa) | 5.61 | 3.08 |

C (–) | 2.60 | 3.26 |

Cold compaction | Hot compaction | |
---|---|---|

A (MPa) | 0 | 0 |

B (MPa) | 5.61 | 3.08 |

C (–) | 2.60 | 3.26 |

*σ*

_{zz}in terms of volumetric plastic strain, which is directly related to the relative density ratio ($\rho \xaf$) through $\epsilon vol=ln(\rho \xaf)$. Finally, the above equation can be updated as follows:

Here, *ρ*_{0} is the initial density of the powder material and *ρ*_{R} is the reference density of the material. Figures 15 and 16 also show the relation between *σ*_{zz} and the relative density ratio $\rho \xaf$ calculated by the analytical solution (Eq. (22)). A good correlation is achieved for both examples. This is a nonlinear behavior, direct extrapolation to high axial pressure regions is difficult. Since the analytical solution is mechanics-based, it helps to understand the mechanism of the powder compaction process.

## 5 Conclusions

This paper studied two test methods that can improve the density ratio of powder compaction. The relative density ratio of the specimen made under the hot compaction method at 300 °C has an obvious increase compared to that of the cold compaction method. The strength of the specimen made from the vibration loading is much better than that of a quasi-static one, although the vibration loading does not obviously improve the relative density. More works are needed to explore under other loading conditions, such as different vibration amplitudes and frequencies in future research.

Drucker–Prager Cap constitutive model was successfully calibrated in abaqus to simulate the powder compaction process. The results of the simulation were correlated very well to test data for the cold compaction and the hot compaction (300 °C) under the 10 kN loading condition.

Finally, an analytical solution for the relative material density during the powder compaction process was developed, which links the axial pressure and the relative density ratio. The results among analytical solutions, test data, and simulation models have a very good match. This analytical solution will give a quick prediction (without numerical simulation) of the material relative densities exceeding testing ranges.

## Acknowledgment

This research work was partially sponsored by the Department of Energy (Grant No. DE-EE0007864) and the University of Central Florida. Thanks to the abaqus (Simulia) software license support from Dassault Systèmes.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment. No data, models, or codes were generated or used for this paper.