Drucker Prager Cap Model Research Articles

Experimental and numerical analysis of structural inhomogeneity of grooved compacts

Structural inhomogeneity developed within compacted powders using punches with grooved surfaces, is a critical issue due the cracking tendency. This study aims to provide a better understanding of the structural inhomogeneity according to the design of the punch surfaces using a finite element modelling. Two case studies are compared. In this modelling, the Drucker-Prager Cap model with density-dependent parameters is considered while the density distributions of manufactured grooved compacts are determined using X-ray tomography. The structural inhomogeneity is then analysed with the goal to determine which punch design is likely to improve the homogeneity.The contact-stress, stress and density distributions within the compact are computed. Predictions of the density distributions are validated from the measurements and show high densified regions localized essentially under the grooves and decreases drastically towards the compact center. The inhomogeneity of structure generated by each punch design is explained by the resulting contact-stress and the wall friction. A notable improvement of the homogeneity is qualitatively showed using the design of five instead of two grooves which was confirmed by X-ray tomography images and modelling thanks to the regular distribution of contact-pressure over the powder bed, even though this last is not uniform.

Determining the Drucker-Prager Cap model constants using experimental, numerical and optimization for compacted Mg powders at different strain rates

This article investigates an inverse approach to determine the coefficients of the Drucker-Prager model for magnesium powder. The approach involves conducting finite element simulations of the powder compression process within LS-DYNA software, employing the Drucker-Prager material model. The goal is to minimize the disparity between force-displacement outcomes derived from simulations and experimental data using a surrogate optimization method. Experimental data were obtained through a uniaxial compression test and served as a basis for adjusting the Cap model coefficients. A random selection of coefficients was made using the Latin cube method and simulations were performed based on the initial coefficients. The optimization was then performed using the particle swarm algorithm over 20 iterations. The optimized coefficients were validated against experimental data, demonstrating close agreement. By utilizing the extracted coefficients, the relative density of the samples was calculated at three different compaction speeds, i.e., 15.5 m s−1 (using a Hopkinson bar), 8 m s−1 (using a drop weight), and 1 mm min−1 (using an Instron machine). The analysis revealed the highest relative density and stress in the densified sample via the Hopkinson bar method, reaching 99.83% and 1.1 GPa, respectively.

Density dependent constitutive model for Bi-2212 powder compression deformation

Bi-2212 HTS materials are fabricated into multi-filamentary wires via powder-in-tube (PIT) method followed by proper heat treatment to obtain superconductivity, but how to predict the large compression deformation behaviors of the Bi-2212 powder is critical to design the processing of the Bi-2212 HTS wire. Drucker Prager/Cap (DPC) model was the most commonly used model for powders including Bi-2212 with soil-like mechanical behavior to consider its shear failure as well as hydrostatic compression. However, the parameters for DPC Cap evolve with densities change and the original model is inadequate to precisely describe the densification process of Bi-2212 powder with large strain. In this study, the modified DPC model with density dependent parameters was introduced for Bi-2212 powder compressions by measuring the failure strength and hydrostatic compressive behavior under different density states. The DPC yield surface was plotted with an evolution trend of non-linear outward expansion with density increased. FEM model of uniaxial compression based on the as-introduced model was built with subroutine VUSDFLD applied. The distribution of Mises stress and relative density were analyzed. The axial stress-density curve for FEM and experimental results were normalized and quantitively evaluated by Mean Square Error (MSE). The introduced model shows good convergence and could match the experimental results well with normalized MSE of 0.000207 and Root Mean Square Error (RMSE) of 0.0144, indicating the mean error percentage of 1.44%. The model introduced in this article provides supports toward large strain deformation simulation of Bi-2212 powder.

An enhanced DPC model for metal powder incorporating yield strength of solid-state

The Drucker-Prager/Cap (DPC) Model has a significant drawback in metal powder metallurgy simulations, which is the potential for the von Mises stress to exceed the yield strength of the solid state. To address this issue, this paper introduces a novel hardening law that effectively restricts the von Mises stress within the yield strength of solid alloy. As the powder becomes denser, the yield surface of the DPC model gradually approach towards the von Mises yield surface. To validate the proposed model, both micro-scale finite element simulations and published experimental data were used. The effectiveness of the proposed model was demonstrated through its application in the simulation of two scenarios: cold die compaction of aluminum powder and hot isostatic pressing of a Nickel based superalloy. The results indicate that the proposed model outperforms the original DPC model and is more suitable for powder metallurgy simulations.

Novel method for the accurate calculation of Drucker–Prager cap model parameters and reduction of experimental time and effort

The Drucker–Prager cap (DPC) model is useful for estimating stress and strain distributions in compacted powders; however, it requires numerous stress state parameters, which are obtained from various tests, thus requiring significant time and effort, and a large number of samples. In addition, there is room for improvement in the reproducibility of DPC model simulations with respect to realistic behavior. This study develops a simple method for estimating DPC model parameters, and it can accurately reproduce the pressures measured during tableting. This objective is achieved by estimating the onset of particle deformation using the powder compaction equation and including the shear stress that results from die wall friction in the stress tensor.

Inverse Identification of Drucker–Prager Cap Model for Ti-6Al-4V Powder Compaction Considering the Shear Stress State

Numerical simulation is an important method to investigate powder-compacting processes. The Drucker–Prager cap constitutive model is often utilized in the numerical simulation of powder compaction. The model contains a number of parameters and it requires a series of mechanical experiments to determine the parameters. The inverse identification methods are time-saving alternatives, but most procedures use a flat punch during the powder-compacting process. It does not reflect the densification behavior under a shearing stress state. Here, an inverse identification approach for the Drucker–Prager cap model parameters is developed by using a hemispherical punch for the powder-compacting experiment. The error between the numerical and experimental displacement–load curves was minimized to identify the Drucker–Prager cap model of titanium alloy powder. The identified model was then verified by powder-compacting experiments with the flat punch. The displacement–load curves acquired by numerical simulation were compared to the displacement–load curves obtained through experiments. The two curves are found to be in good agreement. Meanwhile, the relative density distribution of the powders is similar to the experimental results.

Role of the temperature and aging in mechanical modeling of the active coating in Li-ion battery

Safety of lithium-ion batteries under mechanical loading poses a significant and urgent challenge in the Electric Vehicle (EV) industry. To assess the safety tolerance of the entire battery system, it is crucial to model the batteries subjected to mechanical abuse. The mechanical behavior of batteries is affected by temperature and aging, leading to substantial changes in the properties of active layers. Ignoring these factors may lead to an incomplete estimation of the batteries’ mechanical response.This study examines the results of component tests conducted on batteries at various temperatures and states of health (SOH). The analysis reveals that the particle and adhesion aspects contribute independently to the temperature effect and the aging effect. By incorporating the mechanical interpretation of parameters in the Drucker-Prager Cap (DPC) model, a methodology for characterizing the mechanical properties of in-situ active coatings under different aging conditions and temperatures is introduced. Additionally, the formulation of temperature effects on batteries at different SOH levels is presented. The comparison between finite element (FE) simulations and component tests further confirms the validity of the engineering relationship.

Optimal Performance of Mg-SiC Nanocomposite: Unraveling the Influence of Reinforcement Particle Size on Compaction and Densification in Materials Processed via Mechanical Milling and Cold Iso-Static Pressing

Achieving uniformly distributed reinforcement particles in a dense matrix is crucial for enhancing the mechanical properties of nanocomposites. This study focuses on fabricating Mg-SiC nanocomposites with a high-volume fraction of SiC particles (10 vol.%) using cold isostatic pressing (CIP). The objective is to obtain a fully dense material with a uniform dispersion of nanoparticles. The SiC particle size impact on the compressibility and density distribution of milled Mg-SiC nanocomposites is studied through the elastoplastic Modified Drucker-Prager Cap (MDPC) model and finite element method (FEM) simulations. The findings demonstrate significant variations in the size and dispersion of SiC particles within the Mg matrix. Specifically, the Mg-SiC nanocomposite with 10% submicron-scale SiC content (M10Sµ) exhibits superior compressibility, higher relative density, increased element volume (EVOL), and more consistent density distribution compared to the composite containing 10% nanoscale SiC (M10Sn) following CIP simulation. Under 700 MPa, M10Sµ shows improvements in both computational and experimental results for volume reduction percentage, 2.31% and 2.81%, respectively, and relative density, 4.14% and 3.73%, respectively, compared to M10Sn. The relative density and volume reduction outcomes are in qualitative alignment with experimental findings, emphasizing the significance of particle size in optimizing nanocomposite characteristics.

Prediction of Critical Quality Attributes Based on the Numerical Simulation of Stress and Strain Distributions in Pharmaceutical Tablets.

Various stresses and strains are generated on the surface and inside of pharmaceutical tablets when an external force is applied. In addition, stresses in various directions can remain on the surface and inside the tablets because they are generally prepared by compaction of pharmaceutical powders using dies and punches. As it is difficult to measure the stress and strain generation in the tablets experimentally, a numerical simulation was applied by employing a finite element method (FEM). An elastic model is often used to represent stress and strain generation after loading an external force to tablets, and the Drucker-Prager cap (DPC) model has been widely recognized for representing the remaining stress distributions during the compaction of powder to tablet form. Firstly, this article describes an FEM simulation of the stress generation on the surface of the scored tablets after loading the bending force from the back side of the tablets. Next, the FEM simulation was introduced to determine the effect of diametrical compression on the stress and strain generation in the tablets by comparing the results measured experimentally. Furthermore, the residual stresses remaining inside the tablets were simulated using FEM, in which powder compaction was represented as the DPC model. A clear difference was observed in the residual stress distributions between the flat and convex tablets. This indicates that FEM simulation is useful for achieving a science-based understanding of critical quality attributes in various types of tablets.

Numerical Simulation of Physical Fields during Spark Plasma Sintering of Boron Carbide.

Spark plasma sintering is a new technology for preparing ceramic materials. In this article, a thermal-electric-mechanical coupled model is used to simulate the spark plasma sintering process of boron carbide. The solution of the thermal-electric part was based on the charge conservation equation and the energy conservation equation. A phenomenological constitutive model (Drucker-Prager Cap model) was used to simulate the densification process of boron carbide powder. To reflect the influence of temperature on sintering performance, the model parameters were set as functions of temperature. Spark plasma sintering experiments were conducted at four temperatures: 1500 °C, 1600 °C, 1700 °C, and 1800 °C, and the sintering curves were obtained. The parameter optimization software was integrated with the finite element analysis software, and the model parameters at different temperatures were obtained through the parameter inverse identification method by minimizing the difference between the experimental displacement curve and the simulated displacement curve. The Drucker-Prager Cap model was then incorporated into the coupled finite element framework to analyze the changes of various physical fields of the system over time during the sintering process.

The distribution of Drucker-Prager Cap model parameters for pharmaceutical materials

A model of the compaction process that has come into increasing use with the pharmaceutical industry is the Drucker-Prager Cap model (DPC). It has long been used in conjunction with the finite element methods to simulate the density and stress distributions in compacts. More recently the calibrated parameters that make up the model have been used to characterize the material properties of many compounds. This work will summarize the findings from the collection of DPC parameters from 60 materials. From this dataset several important conclusions can be drawn about typical and extreme behaviors of formulations and excipients. From these findings key signals of program risk can be determined and utilized early in the formulation development process to avoid costly manufacturing and scale-up issues.

On the thermodynamic consistency of the Drucker–Prager Cap model: Modification of the plastic potential and analysis of unloading

The Drucker–Prager Cap (DPC) model is widely used to simulate powder compaction, especially in the pharmaceutical field. However, it has long been recognized that the model may predict an excessive dilation during unloading, most notably for small values of cohesion and/or friction angle. In this article, it is shown that this phenomenon is related to an inherent inconsistency between the yield surface and the commonly used plastic potential. As a result, thermodynamically inconsistent results may be obtained for some parameter values. A modified plastic potential is proposed that guarantees thermodynamic consistency. Uniaxial unloading of a homogeneous tablet in a die with constant cross section and negligible friction is considered so that the elastoplastic equations admit exact closed-form analytical solutions. Hence, exact expressions for the dependence of the axial and radial pressure on tablet height during unloading are obtained. These can be used to analyse experimental unloading data.

Numerical Analysis for Tableting Process in Consideration of Compression Speed

A numerical study of the tableting process using a finite element method (FEM) is important to quantitatively understand the structural change inside the tablet and the mechanism of tableting failures such as capping, picking, lamination, and sticking. In the pharmaceutical field, the Drucker-Prager Cap (DPC) model is used most widely to demonstrate the mechanical behavior of the powder during tableting. The DPC model, however, cannot consider compaction speed, although the compaction speed has a large impact on the tablet strength and tableting failures. In our present study, a DPC–Perzyna model, which incorporates a visco-plastic behavior considering the compression speed, was proposed. Here, the DPC–Perzyna model was outlined and typical results obtained by the DPC–Perzyna model was presented.

Thermo-hydro-mechanical behavior of a clayey rock: A constitutive approach and numerical validation

Temperature strongly affects the thermo-hydro-mechanical properties of clayey rocks. Using the Drucker–Prager Cap model and available experimental results, an advanced constitutive model is proposed to describe the influence of temperature on the hydro-mechanical properties of clayey rocks. To replicate the stress-strain behavior of clayey rock under the thermo-hydro-mechanical conditions, transverse isotropic properties are considered in the constitutive model. Attention is particularly focused on the stress-strain behavior of clayey rock at the plastic stage, the evolution laws of cohesion, friction angle, and pre-consolidation pressure are established by introducing factors such as elastic damage, plastic damage and hardening, and thermal damage. A hydraulic conductivity evolution law is presented, accounting for the thermal effect on water properties and porosity. Theoretical studies are implemented in the finite element method software ABAQUS FEA using the subroutine USDFLD. Some theory parameters are verified through back analysis. Finally, the experimental results and numerical simulations are compared, showing the superiority of the established theoretical model.

Predictive modelling of powder compaction for binary mixtures using the finite element method

Despite the widespread use of solid-form drug delivery within the pharmaceutical industry, tablets remain challenging to formulate because their properties depend strongly on the powder composition and details of the compaction process. Powder compaction simulations, using the finite element method (FEM) in combination with the density-dependent Drucker-Prager Cap model, can be used to aid the design process of pharmaceutical tablets. Parametrisation is typically carried out manually and requires experimental data for each powder considered. This becomes cumbersome when considering different ratios of component powders. An automated parameterisation workflow was developed and validated using experimental powder mixtures of micro-crystalline cellulose and dibasic calcium phosphate dihydrate. FEM simulations reproduced experimental compaction curves with a mean error of 2.5% of the maximum compaction pressure. Moreover, a mixing methodology was developed to estimate parameters of mixtures using only pure-component parameters as input. The experimental compaction curves of mixtures were predicted with a mean error of 4.8%.

One- and two-dimensional finite element modelling of thaw consolidation

Coupled thermo-hydro-mechanical finite element (FE) modelling of thaw consolidation is presented. One-dimensional FE analyses are performed for thaw consolidation of a soil column due to self-weight and with a combination of self-weight and surcharge, with the linear and nonlinear void ratio – effective stress – hydraulic conductivity relationships of thawed soil. The nonlinear behaviour of thawed soil is modelled using a modified Drucker–Prager cap model, while the hydraulic conductivity is varied with the void ratio. Finally, two-dimensional FE modelling of thaw consolidation around a warm pipeline buried in permafrost is performed. The rapid reduction of the void ratio with consolidation, especially at the low-stress level, results in a wide variation of hydraulic conductivity within the thawed zone. The significantly large hydraulic conductivity of soil elements along the curved thaw front, as compared to that of thaw consolidated soil, causes the flow of water along the thaw front, instead of a vertical flow, as assumed in previous 1D thaw consolidation modelling of buried pipelines.

Modelling powder compaction with consideration of a deep grooved punch

In this work a modelling approach for predicting density distribution in the vicinity of two facing grooves on a parallelepiped compact surface, is developed and results are validated using X-ray tomography. An experimental hybrid procedure is proposed to calibrate Drucker-Prager Cap model material parameters which are numerically validated and considered for FEM compaction simulation of grooved specimen using Arbitrary Lagrangian Eulerian method (ALE). Results of the predicted density show high values under the groove and low values on its flanks and shoulders. Similar findings were observed in the literature. Additionally, a strong density gradient between the facing grooves is predicted and validated, demonstrating that the calibrated model achieved good agreements with the measurements.The proposed hybrid calibration procedure could be used for other shape and size die not benefiting from radial instrumentation. Moreover, the ALE approach demonstrated its robustness in solving die powder compaction in presence of strong mesh distortions.

Calibration of Drucker-Prager Cap Constitutive Model for Ceramic Powder Compaction through Inverse Analysis.

Phenomenological plasticity models that relate relative density to plastic strain are frequently used to simulate ceramic powder compaction. With respect to the form implemented in finite element codes, they need to be modified in order to define governing parameters as functions of relative densities. Such a modification increases the number of constitutive parameters and makes their calibration a demanding task that involves a large number of experiments. The novel calibration procedure investigated in this paper is based on inverse analysis methodology, centered on the minimization of a discrepancy function that quantifies the difference between experimentally measured and numerically computed quantities. In order to capture the influence of sought parameters on measured quantities, three different geometries of die and punches are proposed, resulting from a sensitivity analysis performed using numerical simulations of the test. The formulated calibration protocol requires only data that can be collected during the compaction test and, thus, involves a relatively smaller number of experiments. The developed procedure is tested on an alumina powder mixture, used for refractory products, by making a reference to the modified Drucker–Prager Cap model. The assessed parameters are compared to reference values, obtained through more laborious destructive tests performed on green bodies, and are further used to simulate the compaction test with arbitrary geometries. Both comparisons evidenced excellent agreement.

Computational analysis and experimental calibration of cold isostatic compaction of Mg-SiC nanocomposite powders

The consolidation process and density distribution of Mg-SiC nanocomposite powder were studied using computational finite element modeling (FEM) and experimental approaches. Cold isostatic pressing (CIP) was employed for producing fully dense Mg-SiC nanocomposites with a homogeneous distribution of SiC nanoparticles. The elastoplastic modified Drucker-Prager Cap (DPC) model was applied to predict SiC nanoparticle density distribution effects on the milled powder compressibility after the CIP. The FEM results revealed that increasing SiC nanoparticles from 1% (M1Sn) to 10% (M10Sn) volume leads to harder compressibility, increasing the maximum equivalent pressure stress value 717.2 to 737.1 MPa, and decreasing the relative density value 94.63% to 88.67%. The maximum element volume (EVOL) for pure Mg (MM), M1Sn, and M10Sn powders was estimated as 21.69, 20.13, and 15, respectively. The model is validated by comparing finite element simulations with experimental results of relatve density and reduction of volume under 700 MPa cold isostatic pressure for MM, M1Sn, and M10Sn powders. The finite element modeling results of samples after the CIP process were consistent with the experimental results. These results indicate the effectiveness of the modified DPC model and confirmed compaction behavior and relative density of Mg-SiC nanocomposite powders.

Numerical simulation of magnetic pulse radial compaction of W-Cu20 powder with a field shaper

The magnetic field in a magnetic pulse radial compaction process was analyzed in ANSYS/Multiphysical software to determine the electromagnetic force distribution on the driver tube. The node electromagnetic forces were then imported into the structure field as a boundary condition in ABAQUS/Explicit software. A modified Drucker-Prager Cap model was then established to reproduce the compaction behavior of W-Cu20 powder by writing a VUSDFLD subroutine. The Cowper-Symonds constitutive model was used to describe the deformation behavior of the driver tube, the pack tube, and the nylon terminal. Finally, the results predicted by numerical simulation were verified by experiment. The velocity, pressure variation, final distribution of relative density, and relative density uniformity during the magnetic pulse radial compaction process of W-Cu20 powder with a field shaper were predicted by the model, and the effect of field shaper on the relative density was analyzed. The results validate the numerical simulation model of magnetic pulse radial compaction. The magnetic pulse radial powder compaction with a field shaper can significantly increase the compacted compound density with the condition that the inner diameter height of the field shaper is greater than the powder filling height. Although the slit of the field shaper can cause an uneven density distribution after compaction, in the effective range of the field shaper, the density unevenness is minor under the described conditions, less than 4.1%.