Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder

doi:10.21203/rs.3.rs-3875686/v1

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Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder

https://doi.org/10.21203/rs.3.rs-3875686/v1

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published 27 Aug, 2024

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Ultrasonication has widely been used in many industries to develop advanced materials, improve materials behaviors, and enhance mechanical strength to name a few. The present investigation aims to accelerate the densification mechanisms during the hot-pressing process of Ti-6Al-4V powder through high power ultrasonication. A computational study has been developed and implemented to simulate the consolidation behavior, which have then been compared with those experimental data to ensure the simulation accuracy. The constitutive equations including thermoplastic and power law creep models, were extracted at each of the aforesaid stages and applied by FORTRAN software, respectively, in the form of UMAT and CREEP subroutines in the simulation. Finally, the simulation results in relative density-time diagrams and density distribution have been compared with the results of experimental tests. The comparison of the simulation and experimental results shows a maximum error of 6.8 and 2.8% in predicting the densification behavior of hot pressing without and with ultrasonication, respectively. The results show the good accuracy of the simulation in predicting final relative density and density distribution with ultrasonic vibrations.

Powder metallurgy

Hot pressing

Ultrasonic vibration

Stress superposition

Hot powder pressing is suitable for producing parts with complex shapes and desired mechanical properties [1]. Microstructural porosities, voids, and material inhomogeneities are of the most challenging issues in powder hot pressing process being usually left after a consequent sintering process and therefore lower mechanical response is rather unavoidable [2, 3]. To deal with the mentioned problems, a variety of post-treatment techniques such as Hot Isostatic Pressing (HIP) [4], temperature change and phase transformation during pressing [5, 6], low frequency vibrations [7] or high power ultrasonication [8–10], etc. have been employed to boost the efficiency of the process at elevated temperatures. The results reported in literature exhibit that the feasibility of achieving higher densities and finer microstructures without the need of expensive equipment or high pressures (stresses) being required in methods such as HIP [11]. The reason may be attributed to (i) the volumetric phenomenon (stress superposition and acoustic softening) [12–14] and (ii) the reduction of external friction forces, both by the application of high power ultrasonic vibrations with the frequency of about > 20 kHz [15].

Based on the consolidation mechanisms, the constitutive condensation equations are obtained in the analytical method. The density relation (density-time) is derived from the strain rate relation in the mentioned equations. In the finite element simulation of powder metallurgy processes, two methods have been used based on (1) condensation mechanisms and governing equations and (2) the modelling type of material being under deformation.

Consolidation mechanisms and governing equations:

Plastic yield function (Thermoplastic)
Densification mechanism constitutive equations (Viscoplastic)

Material/Part modelling with FEM Software:

Macroscopic constitutive law
Powder-level FE simulation

The analysis of the compression process by the yield function of the porous material depends on the strength of the material; moreover, it is also independent of time and strain rate (thermoplastic relationships). On the other hand, the constitutive equations, including power law creep, diffusion, and grain growth, may depend on time and strain rate (i.e. viscoplastic relationships) [16]. Constitutive plasticity and yield function equations are often employed to model powder cold pressing [17]; however, they have been used in some cases to model the powder hot pressing wherein the final density depends on the temperature and stress, and the operation time has yet to be considered. In these cases, it is not possible to determine the density diagram (density-time), and only the final density and density distribution can be estimated independent of time [18]. In powder-level finite element simulation [18, 19], powder particles are placed adjacent with specific arrangements, and each particle has been separately meshed. Due to low dimensions of the powder particles and a heavy data processing, most of the time, a few powder particles are modeled in 2D or 3D dimensions in the powder-level modelling method. While macroscopic modelling of porous material does not consider the contact between the individual powder particles; as a result, data processing costs are reduced [20, 21].

To date, the volume and surface effects of applying ultrasonic vibrations in manufacturing processes have been simulated namely (1) acoustic softening [22, 23], (2) stress superposition [12, 24], and (3) friction effects [24, 25]. In the simulation based on acoustic softening, the deformation behavior of the material depends on the ultrasonic input parameters such as amplitude or acoustic intensity [22, 26]. In acoustic softening; the primary assumption is that the reduction of material yield strength is proportional to the ultrasonic intensity. In the theory of stress superposition, ultrasonic vibrations are defined as periodic stresses with the specific frequency and amplitude [27]. Surface effects are also often described as changing the type of interactions between the tool and the workpiece or even sometimes as a change in the friction coefficient value.

Our previous research [8, 11], applied high power ultrasonic vibrations to improve the forming conditions in the vertical hot pressing of AA1100 and Ti-6Al-4V powder in an attempt to replace it with that of costly HIP process [28]. The present study is aimed to predict the hot consolidation behavior of Ti-6Al-4V alloy powder under high power ultrasonic vibrations. To this end, the dominant densification mechanisms and their constitutive equations have first been extracted. Due to the complexity of the hot condensation constitutive mechanisms as well as the effect of ultrasonic vibrations, the ultrasonic-assisted condensation behavior has been done by combining analytical equations and writing subroutines in ABAQUS. Then, UMAT/CREEP subroutines were created in FORTRAN software to be combined with ABAQUS. The input parameters of the simulation were selected based on the experimental parameters. Ultrasonic assisted hot pressing of Ti-6Al-4V powder was done in 10 min at 750–950ºC under a uniaxial pressure (stress) of 10-30MPa. Then, the simulation results of the finite elements were extracted in the form of a density (density-time) diagram and density distribution in the sample cross-section; the study is under different temperatures and stress conditions in two states of without and with ultrasonic vibrations.

Hot pressing test of the Ti-6Al-4V alloy powders was performed in the two cases of with and without applying ultrasonic vibrations. In this study, a pre-alloyed Ti-6Al-4V titanium alloy powder with a spherical shape, an average particle size of 20 µm, and a maximum particle size of 45 µm was used. A 3 gr cylindrical sample with both diameter and height of 10mm and height was then hot pressed.

Figure 1 shows the experimental setup and procedure of ultrasonic assisted hot pressing of Ti-6Al-4V powders [11, 29]. In this configuration, using an ultrasonic stack assembly with a nominal power of 1kW and resonance frequency of 25 kHz (Fig. 1-b, c), ultrasonic waves with longitudinal modes was applied from above to the powder pressing area. Also, the amplitude was as 5 ± 1 µm being measured by a gap sensor (Pu-05).

The hot-pressing tests of Ti-6Al-4V spherical powders (Fig. 1-d) have been performed based on an isothermal test condition at a temperature of 750, 850, and 950ºC and under the pressure of 10, 20, and 30MPa without and with ultrasonic vibrations for 10 minutes in the Ref [29]. All the tests were carried out in the controlled environment of neutral argon gas and by a floating mold assembly with the ability to apply double-sided pressure on the powder compact (Fig. 1-b). The final density of the fabricated samples was measured using the standard Archimedes test method. To draw the powder densification curve (relative density to time) (Fig. 1-e), the instantaneous height of the sample was determined. For measuring the density distribution, a hardness test was employed.

In a hot pressing operation, the main stages of condensation are as follows: (1) particle rearrangement (relative movement of all particles), (2) plastic deformation and particle fracture, (3) creep of dislocations, and (4) grain boundary diffusion [16, 28]. The effect of the mentioned mechanisms may depend on parameters like the shape and size of the particles, relative density, material strength, operation temperature, and stress imposed on the material, and diffusion and creep coefficients. To predict the material behavior at high temperatures, the governing consolidation mechanisms are to be determined in different test conditions. The main characteristics of the hot condensation of metal powders can be presented as deformation mechanism maps [16, 30]. In the hot pressing operation, the mechanism diagrams are displayed in the form of relative density-ratio of effective stress to yield stress at a constant temperature [5].

Plastic deformation (yielding) is the main condensation factor at the beginning of a hot pressing operation due to the high stress between the powder particles. Increasing the density and reducing the effective stress below the material’s yield stress hinders thermoplastic deformation, and time-dependent phenomena such as creep and diffusion promote densification [6, 19]. In the operating stress range of 10-30MPa and isothermal temperature of 750–950ºC, with the increase of relative density and leaving the plastic yielding regime, the time-dependent creep of dislocations may be the dominant consolidation mechanism. Based on this, the power law creep governing the creep mechanism has been used in both analytical modelling and finite element simulation. Considering the two main mechanisms of thermoplastic deformation and power law creep, the constitutive equations can be described. In finite element simulation, the final relative density results obtained from the thermoplastic model have been used as the initial density in the power creep model.

3.1. Thermoplastic condensation

In a porous material, effective Misses stress and hydrostatic stress cause deformation. By default, ABAQUS uses the Gurson-Tvergard relation for the thermoplastic modelling of porous materials. This model has an acceptable accuracy at a relative density above 0.9 and is often used to model the creation and growth of porosity in fracture simulation. However, it needs to have acceptable accuracy in low relative densities. In thermoplastic modelling, the Fleck plastic yield function (Eq. 1) [20, 31] has been selected as the constitutive model in the hot thermoplastic condensation of Ti-6Al-4V alloy powder wherein ${\sigma }_{m}$ is the average (hydrostatic) stress, ${\sigma }_{e}$ the effective stress, and ${P}_{y}$ the yield strength of the porous material under hydrostatic pressure (Eq. 2). ${P}_{y}$ Depends on the matrix’s yield strength (${\sigma }_{m}$) and residual porosity in the material where $D$ and ${D}_{0}$ are respectively the relative density and initial relative density.

(1)$\varnothing \left(\sigma ,D\right)={\left[\frac{\sqrt{5}{\sigma }_{m}}{3{P}_{y}}\right]}^{2}+{\left[\frac{5{\sigma }_{e}}{18{P}_{y}}+\frac{2}{3}\right]}^{2}-1=0$

$${P}_{y}=2.97.{D}^{2}\frac{\left(D-{D}_{0}\right)}{{D}_{0}}{\sigma }_{m}$$

3.2. Power Law Creep Consolidation

Duva and Crow [32] introduced constitutive equations for consolidating the reinforced materials by power law creep. Eq. 3 is the constitutive equation to calculate the strain rate tensor resulting from the effective and hydrostatic stresses [33]; S represents the effective effective stress (Eq. 4) for the porous material; ${\sigma }_{m}$, $\stackrel{´}{\sigma }$, and ${\sigma }_{e}$ are the hydrostatic, equivalent deviatoric, and effective stresses, respectively. Eq. 5 shows the relations for calculating coefficients a and b [20, 21] for hot pressing of Ti-6Al-4V spherical powder.

$\dot{\epsilon }=A{S}^{n-1}\left[\frac{3}{2}a\stackrel{´}{\sigma }+\frac{1}{3}b\varvec{I}{\sigma }_{m}\right]$	(3)
$S=a{\sigma }_{e}^{2}+b{\sigma }_{m}^{2}$	(4)
$a=1+0.6{\left[\frac{1-D}{D-{D}_{0}}\right]}^{0.87} , b=0.29{\left[\frac{1-D}{D-{D}_{0}}\right]}^{0.78}$	(5)

By determining the components of the strain rate tensor from Eq. 3, the dilatation rate (${\dot{\epsilon }}_{kk}$) and the densification rate ($\dot{D}$) can be determined by Equations 6 and 7, respectively.

$${\dot{\epsilon }}_{kk}={\dot{\epsilon }}_{xx}+{\dot{\epsilon }}_{yy}+{\dot{\epsilon }}_{zz}$$

$$\dot{D}=-D{\dot{\epsilon }}_{kk}$$

3.3. Ultrasonic Effects

Applying ultrasonic vibration to the assembly, the tool oscillates with a peak-to-peak vibration amplitude of ${A}_{0}$. The vibration amplitude of the tool end has been measured to be 5µm. The stress resulting from the vibrations is determined by Eq. 9 wherein $\rho$ is the density of the vibrating material, c is the sound speed in the infinite material, and $V$ is the speed of the vibrating element (Eq. 8). The element velocity is obtained by taking the gradient from the equation of the displacement amplitude, putting the values of the vibration amplitude as 5µm and the resonance frequency as 25 kHz, as well as extracting the sound speed in the infinite material and the density of the Ti-6Al-4V bulk material equal to 4987 m/s and 4430 kg/m³, respectively.

$$V={A}_{0}\omega \text{cos}\omega t$$

$${\stackrel{-}{\sigma }}_{Acoustic}=\rho c\stackrel{-}{V}$$

The FE simulation extracted the condensation mechanisms and their constitutive equations in hot pressing operations and the application of ultrasonic vibrations. According to the test conditions and densification mechanism maps, plastic yield and power law creep are the two main condensation mechanisms [5, 21]; hence, the simulation has been performed in two consecutive models based on the theory of thermoplastic yield of porous materials and the model of power law creep. FE simulation in ABAQUS software was performed using UMAT (thermoplastic yield function) subroutine and CREEP (power law creep) routine. The effect of ultrasonic vibrations was applied based on the theory of stress superposition and by applying the average acoustic stress in the simulation. The inverse analysis method was used to determine the friction and creep coefficients. Figure 2 shows an algorithm followed in the finite element simulation of the conventional/ultrasonic hot-pressing process.

In the thermoplastic modelling of the first stage, the deformation of porous material with a relative density of lower than 0.9 (D < 0.9), the equation of Fleck et al. [34] was used. The modified Zhang algorithm [35] was used in the ABAQUS UMAT subroutine. ABAQUS CREEP routine was used for power law creep densification of metal powders with and without ultrasonic vibrations. The CREEP subroutine cannot receive stress tensor components directly from the ABAQUS program. For this purpose, the tensor components are first determined by the USDFLD subroutine, and then transferred to the CREEP subroutine [33].

Figure 3 shows the boundary conditions and geometrical specifications of the hot-pressing model. The initial size of the sample is 10 mm in diameter and 14 mm in height. A quarter of the sample section is modeled due to symmetry.

FE parameters are divided into three categories based on the method of determining them: (1) parameters extracted from sources, (2) parameters determined from tests, and (3) parameters obtained from the inverse analysis method of experimental test results. The inverse analysis method is a standard method in determining process parameters (such as friction coefficient) used by various researchers [25, 36, 37]. Based on the inverse analysis method, the present study determined the parameters of creep constants (𝐴 and 𝑛) and friction coefficient (𝜇). Figure 4 represents the algorithm of the inverse analysis method to determine (a) creep coefficients and (b) friction coefficient. The initial values and limits of the power law creep constants are extracted from the research of Kim and Yang [20, 21], Schuh and Dunand [38], and Karmai and Dunne [39, 40].

Table 1 shows the simulation input parameters. The mechanical properties of Ti-6Al-4V alloy are derived from the Kim and Yang [20, 21] in terms of temperature. These data are used in hot compaction simulation with and without ultrasonic vibrations. Vibration amplitude and resonant frequency have been measured in the experimental test phase [41], and the sound speed and density have been extracted from [42]. Power law creep constants and friction coefficient were extracted from the inverse analysis. It should be noted that the simulation parameters given in Table 1 are considered the same for both operations without and with ultrasonic vibrations.

Table 1

Parameter values and equations used in the FE simulation
Elastic Properties
Elastic modulus ($GPa$)		Shear modulus ($GPa$)		Poison's ratio
$104.94-0.052079\times T (℃$)			$104.94-0.052079\times T (℃$)		0.34
Thermoplastic & creep properties
Temperature ($℃$)	Flow stress ($MPa$)			Creep exponent			Dorn’s constant ($MPa/s$)
750	$\stackrel{-}{\sigma }=75.60+121.30{\left({\stackrel{-}{\epsilon }}_{m}^{p}\right)}^{0.3572}$			3.20		$1.5\times {10}^{-10}$
850	$\stackrel{-}{\sigma }=20.70+87.65{\left({\stackrel{-}{\epsilon }}_{m}^{p}\right)}^{0.3950}$			2.80		$1\times {10}^{-8}$
950	$\stackrel{-}{\sigma }=9.18+23.04{\left({\stackrel{-}{\epsilon }}_{m}^{p}\right)}^{0.3423}$			2.94		$4\times {10}^{-8}$
Ultrasonic constants
Vibration amplitude		Ultrasonic frequency		Ti-6Al-4V density				Sound speed
$5 \mu m$		25$kHz$		4430$kg/{m}^{3}$				4987$m/s$
Contact/Friction condition
Coefficient of Friction		Model
0.1		Coulomb sliding friction model

5.1. Thermoplastic Simulation

In the thermoplastic simulation, the final relative density of the samples was determined using the implementation of the UMAT subroutine and ABAQUS software as shown in Fig. 5. Since the thermoplastic hot-pressing process is time-independent, only the final relative density of the sample was extracted. The initial relative density value in thermoplastic simulation is extracted from the experimental test results (pre-pressing in ambient temperature) [11]. The simulation results showed that the density value linearly increases with the increase of temperature and stress of the operation in both cases of without and with ultrasonic vibrations. The obtained density values in the thermoplastic simulation are the initial density in the power law creep densification simulation.

Increasing the pressure, the resulting density linearly increases in both cases of with and without ultrasonic vibrations. Further, owing to the addition of acoustic stress to the static forming stress (stress superposition), the density values obtained from the ultrasonicated samples were higher than those conventionally hot pressed.

5.2. Power Law Creep Results

Figure 7. Comparisons between finite element simulation (Sim) and experimental results (Exp) based on Relative density-Time curve at 850°C during conventional and ultrasonic Ti-6Al-4V hot pressing

A comparison between the experimental and simulation results exhibits the accuracy in predicting density values. The simulation results at 950°C were obtained at the beginning of the compression time in both cases without and with ultrasonic vibrations less than the experimental test values. However, at the end of the operation time, it was close to the experimental values. At 850°C, the simulation and experimental results are consistent with the application of ultrasonic vibrations. While in the case without applying ultrasonic vibrations, with increasing operating pressure (stress) from 10 to 30MPa, the predicted values also increased, so that at 10MPa pressure, the simulation results were less, and at 30MPa pressure, the simulation results were more than the experimental density diagram.

Figure 9. Interaction of temperature and pressure effects in final relative density results from finite element simulation of Ti-6Al-4V alloy powder hot pressing (a) without and (b) with ultrasonic vibrations

The most important sources of error in FE simulation are as follows: selected constitutive models and equations, element size and type, initial test conditions (initial density), material properties (stress-strain diagram at high temperature and creep constants) and, friction conditions and friction model. To investigate the effect of element size and type, mesh sensitivity analysis was performed, and in different conditions, in terms of element type and size, the same results were obtained with appropriate accuracy. To model the material properties in the thermoplastic densification simulation, the fully densified material stress-strain diagram of the Ti-6Al-4V alloy powder in the sources [20, 21] was used, which may be different from the present powder. In the power creep model simulation, the average coefficients at three pressures of 10, 20, and 30MPa were obtained by inverse analysis at different test conditions to determine the creep constants. It should be noted that the creep constants were obtained from the average of three tests at various pressures; the error of non-repeatability of experimental results is one of the errors in determining the creep constants and, consequently, in predicting the simulation results.

5.3. Density Distribution Results

The density distribution in the sample section at the end of the creep modelling stage is extracted from the ABAQUS software. The simulation results of density distribution in the two cases of without and with ultrasonic vibrations are compared in Table 2. To quantify the comparison of simulation results and experimental test [29], equations 10 and 11, respectively, have presented the method of calculating the simulation error in predicting the final relative density and the density distribution in the sample section. As shown in Table 2, the maximum deviation of the final relative density prediction in the two cases of without and with ultrasonic vibrations are respectively 6.8% and 2.8%. Also, the maximum deviation in density distribution (between the experimental results and finite element simulation) in the two cases of without and with ultrasonic vibrations is equal to 5.7% and 3.7%, respectively, being an acceptable error for predicting the behavior of the density distribution in the cross-section of the samples. Also, these results indicate that the presented model has provided a more accurate prediction in the application conditions of ultrasonic vibrations.

$${D}_{Error}\left(\%\right)=\frac{{D}_{EXP}-{D}_{FEM}}{{D}_{EXP}}\times 100$$

$${\varDelta D}_{Error}\left(\%\right)=\frac{{\varDelta D}_{EXP}-{\varDelta D}_{FEM}}{{\varDelta D}_{EXP}}\times 100$$

Table 2

Simulation error table: Simulation results relative to the experimental final relative density values
Ultrasonic			Conventional
30	20	10	30	20	10	Pressure (MPa)
						Temperature ($℃$)
0.3	0.5	2.8	3.5	6.8	3.0	750	${D}_{ERR}\left(\%\right)$
2.1	1.0	0.5	4.1	1.5	5.2	850
0.5	0.4	1.7	1.9	1.9	1.7	950
0.3	0.1	-	1.2	0.5	-	750	$\varDelta {D}_{ERR}\left(\%\right)$
1.9	3.7	1.5	3.5	1.4	-	850
0.6	0.4	0.2	0.1	0.3	5.7	950

Figure 10 compares the density distribution between the experimental results (EXP) and the FE simulation of the hot-pressed sample at the temperature of 950ºC, under the pressure of 30MPa in (a) without (C) and (b) with (UT) ultrasonic vibrations.

The difference in density distribution resulting from finite element simulation and the results of experimental tests can be attributed to various factors. In modelling friction conditions, the errors such as choosing the friction model, determining the friction coefficient (Inverse analysis), and changing the friction conditions may cause errors in predicting the density distribution of the final sample. Along with the Coulomb sliding friction model, the shear friction model can also be considered, especially in high-temperature conditions where the material's shear strength decreases significantly. By choosing the shear friction model, the shear strength coefficients should be measured in the two states of without and with ultrasonic vibrations by designing and building an arrangement to measure the friction and shear force of the material at high temperatures. While in determining the friction coefficient, the inverse analysis method has been used, comparing the density distribution in the simulation with the experimental density distribution.

Power law creep parameters (Creep and Dorn’s constant), stress-strain diagram at high temperatures, and friction coefficients between powder, tool, and mold are among the most influential inputs of the finite element analysis process. In the present study, the mentioned parameters were extracted through the inverse analysis of the hot pressing process. This method depends on the number of repetitions of hot-pressing tests, repeatability of test results, and test conditions. Although conducting the relevant tests is costly, it helps achieve more accurate results in predicting the material's behavior in the finite element simulation.

It seems, the higher accuracy of predicting the results with ultrasonic vibrations is due to the higher repeatability of the experimental tests, compared to the conventional hot pressing method without ultrasound. One of the most critical factors of non-repeatability is the difference in the initial arrangement of the particles, and the friction constants between the powder and the mold.

In this research, a finite element simulation of applying ultrasonic vibrations in the hot pressing of Ti-6Al-4V spherical powder was done with the help of CREEP and UMAT subroutine in ABAQUS software. In the finite element simulation section, consolidation mechanisms and their constitutive equations in hot pressing and applying ultrasonic vibrations were extracted. According to the test conditions and condensation mechanism maps, plastic yielding and creep are the two main condensation factors. Based on this, the simulation has been carried out in two consecutive models based on the theory of thermoplastic yield function of porous materials and the power law creep model. Finite element simulation was done in ABAQUS software using the UMAT (Thermoplastic yielding) subroutine and CREEP (Power law creep) routine. The effect of ultrasonic vibrations was applied based on the theory of stress superposition and by determining the average acoustic stress in the simulation. The inverse analysis method determined the friction and power law creep coefficients. Examining the simulation results of the finite elements, including the densification diagram, final relative density, and density distribution in the sample section, indicates the appropriate accuracy of the used models and the determined coefficients in predicting the densification behavior during hot pressing operation with and without ultrasonic vibrations. The maximum error in predicting the final relative density without and with ultrasonic vibrations is as 6.8% and 2.8%, respectively. Also, the maximum error in predicting the density distribution without and with the application of ultrasonic vibrations is as 5.7% and 3.7%.

Research data for this article

The authors confirm that the data supporting the findings of this study are available within the article.

Declaration of Competing Interest

The authors declare that the research was conducted without any commercial or financial relationships that could be construed as a potential conflict of interest.

Author Contribution

Rezvan Abedini wrote the main manuscript text. All authors reviewed the manuscript.

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No competing interests reported.

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published 27 Aug, 2024

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Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder

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Journal Publication

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Abstract

Figures

1. Introduction

2. Experiment

3. Theory

3.1. Thermoplastic condensation

3.2. Power Law Creep Consolidation

3.3. Ultrasonic Effects

4. Finite Element Simulation

5. Results and Discussion

5.1. Thermoplastic Simulation

5.2. Power Law Creep Results

5.3. Density Distribution Results

6. Conclusion

Declarations

Declaration of Competing Interest

Author Contribution

References

Additional Declarations

Status:

Journal Publication

Version 1

\(\dot{\epsilon }=A{S}^{n-1}\left[\frac{3}{2}a\stackrel{´}{\sigma }+\frac{1}{3}b\varvec{I}{\sigma }_{m}\right]\)	(3)
\(S=a{\sigma }_{e}^{2}+b{\sigma }_{m}^{2}\)	(4)
\(a=1+0.6{\left[\frac{1-D}{D-{D}_{0}}\right]}^{0.87} , b=0.29{\left[\frac{1-D}{D-{D}_{0}}\right]}^{0.78}\)	(5)

Elastic Properties
Elastic modulus (\(GPa\))		Shear modulus (\(GPa\))		Poison's ratio
\(104.94-0.052079\times T (℃\))			\(104.94-0.052079\times T (℃\))		0.34
Thermoplastic & creep properties
Temperature (\(℃\))	Flow stress (\(MPa\))			Creep exponent			Dorn’s constant (\(MPa/s\))
750	\(\stackrel{-}{\sigma }=75.60+121.30{\left({\stackrel{-}{\epsilon }}_{m}^{p}\right)}^{0.3572}\)			3.20		\(1.5\times {10}^{-10}\)
850	\(\stackrel{-}{\sigma }=20.70+87.65{\left({\stackrel{-}{\epsilon }}_{m}^{p}\right)}^{0.3950}\)			2.80		\(1\times {10}^{-8}\)
950	\(\stackrel{-}{\sigma }=9.18+23.04{\left({\stackrel{-}{\epsilon }}_{m}^{p}\right)}^{0.3423}\)			2.94		\(4\times {10}^{-8}\)
Ultrasonic constants
Vibration amplitude		Ultrasonic frequency		Ti-6Al-4V density				Sound speed
\(5 \mu m\)		25\(kHz\)		4430\(kg/{m}^{3}\)				4987\(m/s\)
Contact/Friction condition
Coefficient of Friction		Model
0.1		Coulomb sliding friction model

Ultrasonic			Conventional
30	20	10	30	20	10	Pressure (MPa)
						Temperature (\(℃\))
0.3	0.5	2.8	3.5	6.8	3.0	750	\({D}_{ERR}\left(\%\right)\)
2.1	1.0	0.5	4.1	1.5	5.2	850
0.5	0.4	1.7	1.9	1.9	1.7	950
0.3	0.1	-	1.2	0.5	-	750	\(\varDelta {D}_{ERR}\left(\%\right)\)
1.9	3.7	1.5	3.5	1.4	-	850
0.6	0.4	0.2	0.1	0.3	5.7	950