Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder

Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder Rezvan Abedini, Vahid Fartashvand, Amir Abdullah, Yunes Alizadeh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3875686/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 27 Aug, 2024 Read the published version in Mechanics of Time-Dependent Materials → Version 1 posted 2 You are reading this latest preprint version Abstract 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 Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction 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. 2. Experiment 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. 3. Theory 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) \(\varnothing \left(\sigma ,D\right)={\left[\frac{\sqrt{5}{\sigma }_{m}}{3{P}_{y}}\right]}^{2}+{\left[\frac{5{\sigma }_{e}}{18{P}_{y}}+\frac{2}{3}\right]}^{2}-1=0\) $${P}_{y}=2.97.{D}^{2}\frac{\left(D-{D}_{0}\right)}{{D}_{0}}{\sigma }_{m}$$ 2 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. $${\dot{\epsilon }}_{kk}={\dot{\epsilon }}_{xx}+{\dot{\epsilon }}_{yy}+{\dot{\epsilon }}_{zz}$$ 6 $$\dot{D}=-D{\dot{\epsilon }}_{kk}$$ 7 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/m 3 , respectively. $$V={A}_{0}\omega \text{cos}\omega t$$ 8 $${\stackrel{-}{\sigma }}_{Acoustic}=\rho c\stackrel{-}{V}$$ 9 4. Finite Element Simulation 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. Results and Discussion 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. $${D}_{Error}\left(\%\right)=\frac{{D}_{EXP}-{D}_{FEM}}{{D}_{EXP}}\times 100$$ 10 $${\varDelta D}_{Error}\left(\%\right)=\frac{{\varDelta D}_{EXP}-{\varDelta D}_{FEM}}{{\varDelta D}_{EXP}}\times 100$$ 11 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. 6. Conclusion 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%. Declarations 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. References Upadhyaya, G.S., Powder metallurgy technology . 1997: Cambridge Int Science Publishing. Bolzoni, L., et al., Inductive hot-pressing of titanium and titanium alloy powders . Materials Chemistry and Physics, 2012. 131(3): p. 672–679. 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Zhou, Q., et al., Comprehensive Studies on Hot Compaction and Vibration-Assisted Compaction Tests of Aluminum Powder . Journal of Manufacturing Science and Engineering, 2020. 143(1). Schuh, C. and D.C. Dunand, Non-isothermal transformation-mismatch plasticity: modeling and experiments on Ti–6Al–4V. Acta Materialia, 2001. 49(2): p. 199–210. Carmai, J. and F.P.E. Dunne, Micromechanical Models for Creep in the Consolidation of Composites , in IUTAM Symposium on Creep in Structures , S. Murakami and N. Ohno, Editors. 2001, Springer Netherlands. p. 463–468. Carmai, J. and F. Dunne, Constitutive equations for densification of matrix-coated fibre composites during hot isostatic pressing . International Journal of Plasticity, 2003. 19(3): p. 345–363. Fartashvand, V., R. Abedini, and A. Abdullah, Influence of ultrasonic vibrations on the properties of press-and-sintered titanium. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2022: p. 09544054221078386. Welsch, G., R. Boyer, and E.W. Collings, Materials Properties Handbook: Titanium Alloys . 1993: ASM International. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 27 Aug, 2024 Read the published version in Mechanics of Time-Dependent Materials → Version 1 posted Submission checks completed at journal 23 Jan, 2024 First submitted to journal 18 Jan, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3875686","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":268795543,"identity":"2291e589-125e-4cdd-a7a6-b67f3b1824ef","order_by":0,"name":"Rezvan Abedini","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVRIiWNgGAWjYBCDBAaGxAaGhAoJmIAEHsUoWs4AVbIRrwWIGNsYYFpwA/P29mcSP3fY5fGzJ7c9eDjPInH+/AbGDz8YLPJxaZE5cyBNsvdMcrFkz8N2g8RtEokbjjEwS/YwSFg24NAiIZFw2IC3jTlxw43ENgmwFqDDpIESBrhskZB/2Gz4t60+cT9YyxyJxPltDMy/8WqRYGZ8zNt2OHGDBEhLAxAdY2DDbwtPGuNj2bbjxRJnHrZJJByTMN5wLLHNsscAjxb24w8Ovm2rzuNvT38m+aOmTnZ+8+HDN35U1OHUgg0wNjAwkKRhFIyCUTAKRgE6AABx8FIVyIxdmgAAAABJRU5ErkJggg==","orcid":"","institution":"Iran University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Rezvan","middleName":"","lastName":"Abedini","suffix":""},{"id":268795544,"identity":"fe8402e4-dd1f-4142-9cd4-fd09d1591d1e","order_by":1,"name":"Vahid Fartashvand","email":"","orcid":"","institution":"Alzahra University","correspondingAuthor":false,"prefix":"","firstName":"Vahid","middleName":"","lastName":"Fartashvand","suffix":""},{"id":268795545,"identity":"8025adfc-e42d-4a6b-bb5a-0a3e80137891","order_by":2,"name":"Amir Abdullah","email":"","orcid":"","institution":"Amirkabir University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Amir","middleName":"","lastName":"Abdullah","suffix":""},{"id":268795546,"identity":"1ed8bfc9-b281-40ff-8c1f-e80f9c9ccba8","order_by":3,"name":"Yunes Alizadeh","email":"","orcid":"","institution":"Amirkabir University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Yunes","middleName":"","lastName":"Alizadeh","suffix":""}],"badges":[],"createdAt":"2024-01-18 11:59:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3875686/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3875686/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11043-024-09735-y","type":"published","date":"2024-08-27T15:57:07+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":50188699,"identity":"b3c7fad8-6d17-4dd2-a227-5ef7da4127fb","added_by":"auto","created_at":"2024-01-25 21:37:53","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":625558,"visible":true,"origin":"","legend":"\u003cp\u003eUltrasonic assisted hot pressing test setup and experimental test results: (a) powder compaction setup, (b) Ultrasonic setup, (c) Ultrasonic frequency results, (d) Ti-6Al-4V spherical powders, (e) Densification curves (Relative Density-Time) and, (f) SEM micrograph of fracture.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/2d423b9c779760fa5c7046cb.png"},{"id":50189112,"identity":"95d2e735-544b-4c9f-8e7b-e924fd488c52","added_by":"auto","created_at":"2024-01-25 21:45:53","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":108321,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element simulation algorithm in the hot-pressing process\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/85e73e6b6b81b5bca46c5fcd.png"},{"id":50189113,"identity":"1638d010-134a-4203-8697-f8479bdd37ab","added_by":"auto","created_at":"2024-01-25 21:45:53","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":345812,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Basic components of the model and (b) Symmetric meshing of powder compact\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/25a222653a3fb3cfeaccc142.png"},{"id":50189111,"identity":"41a073be-a345-4f74-be96-4ff1fb238b09","added_by":"auto","created_at":"2024-01-25 21:45:53","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":91891,"visible":true,"origin":"","legend":"\u003cp\u003eThe algorithm of the inverse analysis method to determine (a) the power law creep (A and n) constants and (b) the friction coefficient (μ)\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/05598e183eee7bb9da2664c2.png"},{"id":50188694,"identity":"9d46fd8f-30cf-4c1f-804a-150629be23f8","added_by":"auto","created_at":"2024-01-25 21:37:53","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":88017,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element simulation results from the models of Fleck et al. [34] for the variation of relative density with (a) pressure and (b) temperature during Conventional (C) and Ultrasonic assisted (UT) hot pressing.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/d015174662369b7f66f3b6ce.png"},{"id":50189114,"identity":"e16050c1-43e1-4774-8f3a-3648f0fd4033","added_by":"auto","created_at":"2024-01-25 21:45:53","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":42510,"visible":true,"origin":"","legend":"\u003cp\u003eComparisons between finite element simulation (Sim) and experimental results (Exp) based on Relative density-Time curve at 750°C during conventional and ultrasonic Ti-6Al-4V hot pressing\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/c7cf7406ac0d17e7944a177b.png"},{"id":50188692,"identity":"403287f6-c6c2-43aa-9747-ae05b5c624de","added_by":"auto","created_at":"2024-01-25 21:37:53","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":45117,"visible":true,"origin":"","legend":"\u003cp\u003eComparisons 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\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/ba35c8a7b154c5e13625983f.png"},{"id":50188693,"identity":"5abaf6c6-bbed-4ccc-92fd-8d96cd8d183e","added_by":"auto","created_at":"2024-01-25 21:37:53","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":40502,"visible":true,"origin":"","legend":"\u003cp\u003eComparisons between finite element simulation (Sim) and experimental results (Exp) based on Relative density-Time curve at 950°C during conventional and ultrasonic Ti-6Al-4V hot pressing\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/116a844f75f49d719e813ea3.png"},{"id":50188697,"identity":"44cb9121-3e8a-4f91-923f-8c5344597f64","added_by":"auto","created_at":"2024-01-25 21:37:53","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":397383,"visible":true,"origin":"","legend":"\u003cp\u003eInteraction 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\u003c/p\u003e","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/e28d001fee109a1aee191fe2.png"},{"id":50188701,"identity":"c5718573-123b-4d3c-ae5e-ae318495b077","added_by":"auto","created_at":"2024-01-25 21:37:53","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":163691,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of density distribution in Finite Elements Modelling (FEM) and experimental results (EXP) in hot pressing (10min, 950ºC, and 30MPa Pressure): (a) without (C) and (b) with (UT) ultrasonic vibrations\u003c/p\u003e","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/8c2ee7f4d4fd6bf39b66a2be.png"},{"id":63821136,"identity":"cf84a812-315a-443d-83cd-1b17059de85f","added_by":"auto","created_at":"2024-09-02 16:12:20","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2640549,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3875686/v1/896d761b-7e3b-489f-8a6d-4b991290f98a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eHot powder pressing is suitable for producing parts with complex shapes and desired mechanical properties [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. To deal with the mentioned problems, a variety of post-treatment techniques such as Hot Isostatic Pressing (HIP) [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], temperature change and phase transformation during pressing [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], low frequency vibrations [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] or high power ultrasonication [\u003cspan additionalcitationids=\"CR9\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], 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 [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. The reason may be attributed to (i) the volumetric phenomenon (stress superposition and acoustic softening) [\u003cspan additionalcitationids=\"CR13\" citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] and (ii) the reduction of external friction forces, both by the application of high power ultrasonic vibrations with the frequency of about\u0026thinsp;\u0026gt;\u0026thinsp;20 kHz [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eBased 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.\u003c/p\u003e \u003cp\u003eConsolidation mechanisms and governing equations:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ePlastic yield function (Thermoplastic)\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eDensification mechanism constitutive equations (Viscoplastic)\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eMaterial/Part modelling with FEM Software:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eMacroscopic constitutive law\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ePowder-level FE simulation\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe 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) [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Constitutive plasticity and yield function equations are often employed to model powder cold pressing [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]; 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 [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. In powder-level finite element simulation [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], 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 [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eTo date, the volume and surface effects of applying ultrasonic vibrations in manufacturing processes have been simulated namely (1) acoustic softening [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e], (2) stress superposition [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], and (3) friction effects [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. 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.\u003c/p\u003e \u003cp\u003eOur previous research [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], 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 [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. 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\u0026ndash;950\u0026ordm;C 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.\u003c/p\u003e"},{"header":"2. Experiment","content":"\u003cp\u003eHot 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 \u0026micro;m, and a maximum particle size of 45 \u0026micro;m was used. A 3 gr cylindrical sample with both diameter and height of 10mm and height was then hot pressed.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the experimental setup and procedure of ultrasonic assisted hot pressing of Ti-6Al-4V powders [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. In this configuration, using an ultrasonic stack assembly with a nominal power of 1kW and resonance frequency of 25 kHz (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e-b, c), ultrasonic waves with longitudinal modes was applied from above to the powder pressing area. Also, the amplitude was as 5\u0026thinsp;\u0026plusmn;\u0026thinsp;1 \u0026micro;m being measured by a gap sensor (Pu-05).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe hot-pressing tests of Ti-6Al-4V spherical powders (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e-d) have been performed based on an isothermal test condition at a temperature of 750, 850, and 950\u0026ordm;C and under the pressure of 10, 20, and 30MPa without and with ultrasonic vibrations for 10 minutes in the Ref [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. 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.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e-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.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e-e), the instantaneous height of the sample was determined. For measuring the density distribution, a hardness test was employed.\u003c/p\u003e"},{"header":"3. Theory","content":"\u003cp\u003eIn 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 [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e].\u003c/p\u003e \u003cp\u003ePlastic 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\u0026rsquo;s yield stress hinders thermoplastic deformation, and time-dependent phenomena such as creep and diffusion promote densification [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. In the operating stress range of 10-30MPa and isothermal temperature of 750\u0026ndash;950\u0026ordm;C, 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.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Thermoplastic condensation\u003c/h2\u003e \u003cp\u003eIn 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.\u0026nbsp;1) [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e] has been selected as the constitutive model in the hot thermoplastic condensation of Ti-6Al-4V alloy powder wherein \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{m}\\)\u003c/span\u003e\u003c/span\u003e is the average (hydrostatic) stress, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{e}\\)\u003c/span\u003e\u003c/span\u003e the effective stress, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{y}\\)\u003c/span\u003e\u003c/span\u003e the yield strength of the porous material under hydrostatic pressure (Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e2\u003c/span\u003e). \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{y}\\)\u003c/span\u003e\u003c/span\u003e Depends on the matrix\u0026rsquo;s yield strength (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{m}\\)\u003c/span\u003e\u003c/span\u003e) and residual porosity in the material where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(D\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D}_{0}\\)\u003c/span\u003e\u003c/span\u003e are respectively the relative density and initial relative density.\u003c/p\u003e \u003cp\u003e(1)\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\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\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${P}_{y}=2.97.{D}^{2}\\frac{\\left(D-{D}_{0}\\right)}{{D}_{0}}{\\sigma }_{m}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Power Law Creep Consolidation\u003c/h2\u003e \u003cp\u003eDuva and Crow [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e] introduced constitutive equations for consolidating the reinforced materials by power law creep. Eq.\u0026nbsp;3 is the constitutive equation to calculate the strain rate tensor resulting from the effective and hydrostatic stresses [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]; S represents the \u003cem\u003eeffective\u003c/em\u003e effective stress (Eq.\u0026nbsp;4) for the porous material; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{m}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{\u0026acute;}{\\sigma }\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{e}\\)\u003c/span\u003e\u003c/span\u003e are the hydrostatic, equivalent deviatoric, and effective stresses, respectively. Eq.\u0026nbsp;5 shows the relations for calculating coefficients a and b [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] for hot pressing of Ti-6Al-4V spherical powder.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\dot{\\epsilon }=A{S}^{n-1}\\left[\\frac{3}{2}a\\stackrel{\u0026acute;}{\\sigma }+\\frac{1}{3}b\\varvec{I}{\\sigma }_{m}\\right]\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(3)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S=a{\\sigma }_{e}^{2}+b{\\sigma }_{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(4)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(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}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(5)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eBy determining the components of the strain rate tensor from Eq.\u0026nbsp;3, the dilatation rate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\dot{\\epsilon }}_{kk}\\)\u003c/span\u003e\u003c/span\u003e) and the densification rate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\dot{D}\\)\u003c/span\u003e\u003c/span\u003e) can be determined by Equations \u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e6\u003c/span\u003e and \u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e7\u003c/span\u003e, respectively.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${\\dot{\\epsilon }}_{kk}={\\dot{\\epsilon }}_{xx}+{\\dot{\\epsilon }}_{yy}+{\\dot{\\epsilon }}_{zz}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\dot{D}=-D{\\dot{\\epsilon }}_{kk}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3. Ultrasonic Effects\u003c/h2\u003e \u003cp\u003eApplying ultrasonic vibration to the assembly, the tool oscillates with a peak-to-peak vibration amplitude of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({A}_{0}\\)\u003c/span\u003e\u003c/span\u003e. The vibration amplitude of the tool end has been measured to be 5\u0026micro;m. The stress resulting from the vibrations is determined by Eq.\u0026nbsp;\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e9\u003c/span\u003e wherein \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho\\)\u003c/span\u003e\u003c/span\u003e is the density of the vibrating material, c is the sound speed in the infinite material, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V\\)\u003c/span\u003e\u003c/span\u003e is the speed of the vibrating element (Eq.\u0026nbsp;\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e8\u003c/span\u003e). The element velocity is obtained by taking the gradient from the equation of the displacement amplitude, putting the values of the vibration amplitude as 5\u0026micro;m 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/m\u003csup\u003e3\u003c/sup\u003e, respectively.\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$V={A}_{0}\\omega \\text{cos}\\omega t$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${\\stackrel{-}{\\sigma }}_{Acoustic}=\\rho c\\stackrel{-}{V}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Finite Element Simulation","content":"\u003cp\u003eThe 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 [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]; 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\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows an algorithm followed in the finite element simulation of the conventional/ultrasonic hot-pressing process.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the thermoplastic modelling of the first stage, the deformation of porous material with a relative density of lower than 0.9 (D\u0026thinsp;\u0026lt;\u0026thinsp;0.9), the equation of Fleck et al. [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e] was used. The modified Zhang algorithm [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] 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 [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFE 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 [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e]. Based on the inverse analysis method, the present study determined the parameters of creep constants (\u0026#119860; and \u0026#119899;) and friction coefficient (\u0026#120583;). Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e 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 [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], Schuh and Dunand [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], and Karmai and Dunne [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e, \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the simulation input parameters. The mechanical properties of Ti-6Al-4V alloy are derived from the Kim and Yang [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] 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 [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e], and the sound speed and density have been extracted from [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. Power law creep constants and friction coefficient were extracted from the inverse analysis. It should be noted that the simulation parameters given in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e are considered the same for both operations without and with ultrasonic vibrations.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eParameter values and equations used in the FE simulation\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"9\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"9\" nameend=\"c9\" namest=\"c1\"\u003e \u003cp\u003eElastic Properties\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eElastic modulus\u003c/p\u003e \u003cp\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(GPa\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003eShear modulus\u003c/p\u003e \u003cp\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(GPa\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"5\" nameend=\"c9\" namest=\"c5\"\u003e \u003cp\u003ePoison's ratio\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c3\" namest=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(104.94-0.052079\\times T (℃\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(104.94-0.052079\\times T (℃\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"4\" nameend=\"c9\" namest=\"c6\"\u003e \u003cp\u003e0.34\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"9\" nameend=\"c9\" namest=\"c1\"\u003e \u003cp\u003eThermoplastic \u0026amp; creep properties\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTemperature (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(℃\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003eFlow stress\u003c/p\u003e \u003cp\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(MPa\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c7\" namest=\"c5\"\u003e \u003cp\u003eCreep exponent\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c9\" namest=\"c8\"\u003e \u003cp\u003eDorn\u0026rsquo;s constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(MPa/s\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e750\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{\\sigma }=75.60+121.30{\\left({\\stackrel{-}{\\epsilon }}_{m}^{p}\\right)}^{0.3572}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e3.20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c9\" namest=\"c7\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(1.5\\times {10}^{-10}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e850\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{\\sigma }=20.70+87.65{\\left({\\stackrel{-}{\\epsilon }}_{m}^{p}\\right)}^{0.3950}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e2.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c9\" namest=\"c7\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(1\\times {10}^{-8}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e950\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{\\sigma }=9.18+23.04{\\left({\\stackrel{-}{\\epsilon }}_{m}^{p}\\right)}^{0.3423}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e2.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c9\" namest=\"c7\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(4\\times {10}^{-8}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"9\" nameend=\"c9\" namest=\"c1\"\u003e \u003cp\u003eUltrasonic constants\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eVibration amplitude\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003eUltrasonic frequency\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"4\" nameend=\"c8\" namest=\"c5\"\u003e \u003cp\u003eTi-6Al-4V density\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSound speed\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(5 \\mu m\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003e25\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(kHz\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"4\" nameend=\"c8\" namest=\"c5\"\u003e \u003cp\u003e4430\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(kg/{m}^{3}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e4987\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m/s\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"9\" nameend=\"c9\" namest=\"c1\"\u003e \u003cp\u003eContact/Friction condition\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eCoefficient of Friction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c8\" namest=\"c3\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c8\" namest=\"c3\"\u003e \u003cp\u003eCoulomb sliding friction model\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"5. Results and Discussion","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e5.1. Thermoplastic Simulation\u003c/h2\u003e \u003cp\u003eIn 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.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. 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) [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIncreasing 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.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e5.2. Power Law Creep Results\u003c/h2\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eFigure 7.\u003c/b\u003e Comparisons between finite element simulation (Sim) and experimental results (Exp) based on Relative density-Time curve at 850\u0026deg;C during conventional and ultrasonic Ti-6Al-4V hot pressing\u003c/p\u003e \u003cp\u003eA comparison between the experimental and simulation results exhibits the accuracy in predicting density values. The simulation results at 950\u0026deg;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\u0026deg;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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. 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\u003c/p\u003e \u003cp\u003eThe 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 [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] 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.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e5.3. Density Distribution Results\u003c/h2\u003e \u003cp\u003eThe 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\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. To quantify the comparison of simulation results and experimental test [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e], equations \u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e10\u003c/span\u003e and \u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e11\u003c/span\u003e, 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\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, 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.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${D}_{Error}\\left(\\%\\right)=\\frac{{D}_{EXP}-{D}_{FEM}}{{D}_{EXP}}\\times 100$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${\\varDelta D}_{Error}\\left(\\%\\right)=\\frac{{\\varDelta D}_{EXP}-{\\varDelta D}_{FEM}}{{\\varDelta D}_{EXP}}\\times 100$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSimulation error table: Simulation results relative to the experimental final relative density values\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"12\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c12\" colnum=\"12\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"4\" nameend=\"c4\" namest=\"c1\"\u003e \u003cp\u003eUltrasonic\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"4\" nameend=\"c8\" namest=\"c5\"\u003e \u003cp\u003eConventional\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c11\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colspan=\"1\" nameend=\"c12\" namest=\"c12\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"4\" nameend=\"c12\" namest=\"c9\"\u003e \u003cp\u003ePressure (MPa)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"4\" nameend=\"c12\" namest=\"c9\"\u003e \u003cp\u003eTemperature (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(℃\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e3.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e6.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e3.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003e750\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" morerows=\"2\" nameend=\"c12\" namest=\"c11\" rowspan=\"3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D}_{ERR}\\left(\\%\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e2.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e4.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e5.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003e850\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e1.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003e950\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003e750\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" morerows=\"2\" nameend=\"c12\" namest=\"c11\" rowspan=\"3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varDelta {D}_{ERR}\\left(\\%\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e1.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e3.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003e850\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003e0.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e5.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c10\" namest=\"c9\"\u003e \u003cp\u003e950\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e compares the density distribution between the experimental results (EXP) and the FE simulation of the hot-pressed sample at the temperature of 950\u0026ordm;C, under the pressure of 30MPa in (a) without (C) and (b) with (UT) ultrasonic vibrations.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe 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.\u003c/p\u003e \u003cp\u003ePower law creep parameters (Creep and Dorn\u0026rsquo;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.\u003c/p\u003e \u003cp\u003eIt 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.\u003c/p\u003e \u003c/div\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003eIn 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 \u003cspan refid=\"Sec7\" class=\"InternalRef\"\u003efinite element simulation\u003c/span\u003e 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%.\u003c/p\u003e "},{"header":"Declarations","content":"\u003cp\u003e \u003cb\u003eResearch data for this article\u003c/b\u003e \u003c/p\u003e \u003cp\u003eThe authors confirm that the data supporting the findings of this study are available within the article.\u003c/p\u003e\n\u003cp\u003e \u003ch2\u003eDeclaration of Competing Interest\u003c/h2\u003e \u003cp\u003eThe authors declare that the research was conducted without any commercial or financial relationships that could be construed as a potential conflict of interest.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eRezvan Abedini wrote the main manuscript text. 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Collings, \u003cem\u003eMaterials Properties Handbook: Titanium Alloys\u003c/em\u003e. 1993: ASM International.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"mechanics-of-time-dependent-materials","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mtdm","sideBox":"Learn more about [Mechanics of Time-Dependent Materials](http://link.springer.com/journal/11043)","snPcode":"11043","submissionUrl":"https://submission.nature.com/new-submission/11043/3","title":"Mechanics of Time-Dependent Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Powder metallurgy, Hot pressing, Ultrasonic vibration, Stress superposition","lastPublishedDoi":"10.21203/rs.3.rs-3875686/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3875686/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eUltrasonication 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.\u003c/p\u003e","manuscriptTitle":"Finite Element Modelling of Ultrasonic Assisted Hot Pressing of Metal Powder","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-25 21:37:49","doi":"10.21203/rs.3.rs-3875686/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"checksComplete","content":"","date":"2024-01-23T14:09:11+00:00","index":"","fulltext":""},{"type":"submitted","content":"Mechanics of Time-Dependent Materials","date":"2024-01-18T11:54:55+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"mechanics-of-time-dependent-materials","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mtdm","sideBox":"Learn more about [Mechanics of Time-Dependent Materials](http://link.springer.com/journal/11043)","snPcode":"11043","submissionUrl":"https://submission.nature.com/new-submission/11043/3","title":"Mechanics of Time-Dependent Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"b0b48e46-a67e-447d-b9a4-bfc1ba958f8d","owner":[],"postedDate":"January 25th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-09-02T16:05:59+00:00","versionOfRecord":{"articleIdentity":"rs-3875686","link":"https://doi.org/10.1007/s11043-024-09735-y","journal":{"identity":"mechanics-of-time-dependent-materials","isVorOnly":false,"title":"Mechanics of Time-Dependent Materials"},"publishedOn":"2024-08-27 15:57:07","publishedOnDateReadable":"August 27th, 2024"},"versionCreatedAt":"2024-01-25 21:37:49","video":"","vorDoi":"10.1007/s11043-024-09735-y","vorDoiUrl":"https://doi.org/10.1007/s11043-024-09735-y","workflowStages":[]},"version":"v1","identity":"rs-3875686","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3875686","identity":"rs-3875686","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}