Modified Drucker-Prager Cap Model Research Articles

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.

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.

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%.

Modeling the powder compaction process by an integrated simulation and inverse optimization method

A method of integrated modeling using the inverse optimization was proposed to simulate the compaction process of metal powders in this work. By using the present work, one can alleviate the time consuming and expensive experiment, and the material parameters can be determined more accurately. The constitutive parameters of the modified Drucker-Prager Cap model were identified using finite element simulations in ABAQUS- iSIGHT integrated environment. The downhill simplex optimization method has been used to predict the material parameters in a way to minimize the differences between experimental and simulation result of uniaxial powder compaction. The accuracy of inverse parameters has been validated as the powder Distaloy AE, while the sensitivity of the plasticity parameters of the DPC model has been discussed. The results show that the ejection process cannot be fully simulated when the plastic parameters are assumed to be constants, while for the case that adopting plastic parameters changed with relative density can show a good agreement with the experimental data. Finally, the parametric module of Python was applied to simulate the compaction of industrial part. The integrated modeling using inverse optimization method was verified again in compaction of a typical multi-level gear component. And the numerical results show that the errors of the maximum and the minimum relative density are about 5% and 4% respectively.

Study of Compaction of the Titanium Powder by the Experimental-Analytical Method

The dependence of porosity θ of the spongy titanium-based powder material on stress state coefficient k during the plastic deformation with the prevailing effect of the uniform compression is investigated. Based on the results found in previous works, the family of yield curves with a variable porosity is plotted on the σ–T plane. The yield criterion of the powder material is based on the Modified Drucker–Prager Cap Model. Straight lines corresponding to various values of stress-state coefficient k = σ/T, where σ is the average hydrostatic stress and T is the intensity of tangential stresses, are represented in the graph of the geometric interpretation of the accepted plasticity model. To formulate the relation of porosity (θ, %), average normal stress ( $$\bar {\sigma }$$ ), expressed in the dimensionless form, and stress-state coefficient k, intersection points of the family of curves corresponding to generatrices of yield surfaces on the σ–T plane and radial straight lines, are used. This results in the derivation of the equation of the θ = θ ( $$\bar {\sigma }$$ , k) form. To verify the adequacy of this relationship, the experimental part of the investigation is fulfilled. Powder billets preliminarily compacted at a pressure of 1000 MPa and temperature 325°C are subjected to electroerosion cutting along the axial section to form planar samples (templates). Several characteristic segments are selected on the template surface to determine the local surface porosity by quantitative metallography. The stress state was additionally determined in representative segments by the numerical simulation. Values of bulk plastic strain ( $$\varepsilon _{{v}}^{{{\text{pl}}}}$$ ), intensities of tangential stresses (T), and hydrostatic normal stress (σ) are calculated in axial section zones corresponding to the regions under study. It is shown that the stress state coefficient insignificantly affects the porosity upon its variation in a rather broad range (k = –10 to –0.86).

Sensitivity of important parameters in a three‐dimensional simulation of the milling process of sugar cane with modified Drucker–Prager Cap model based on evolutionary material properties

The properties of the milled mixture of sugar cane change greatly during the milling process. The evolution of its properties was modeled using three-dimensional simulation method of the modified Drucker–Prager Cap model. Parameters for the model were determined by dynamic compression tests and simulated contrast test and employed to analyze the changes of roll load, roll torque, the maximum speed of sugar juice, stress, pore pressure, and the maximum void ratio under compression ratios of 1.5–3.5, blanket thicknesses of 40–140 mm, roll diameters of 700–1000 mm, and roll surface speeds of 0.1–0.5 m/s. The following results have been found: compression ratio plays the leading role in stress, pore pressure, roll load, and roll torque which increase with it; blanket thickness is of primary importance for maximum void ratio which increases with it; and roll surface speed has the most obvious effect on maximum sugar juice speed which increases with it. The method of this paper might provide a more accurate prediction for the optimization of these important parameters during the milling process of sugar cane. Practical applications The properties of the milled mixture of sugar cane change greatly during the milling process. However, the material properties for the constitutive model adopted by the researchers were the mean values of those measured. As a result, the predicted results obtained by the model based on the average values of these parameters are different from those obtained by the actual tests. We evolved the law of variation of parameters and adopted a three-dimensional simulation method of the modified Drucker–Prager Cap model to the milling process of sugar cane. Our results show compression ratio is most important for stress, pore pressure, roll load, and roll torque which increase with it; blanket thickness and roll surface speed are of primary importance for maximum void ratio and the most obvious effect on maximum sugar juice speed which increases with them. Information presented here can serve as a guidance for a more accurate prediction for the optimization of these important parameters during the milling process of sugar cane.

Study on elastoplastic coupling mechanics model of the milled mixture of sugar cane

Abstract At present, the phenomenon of elastoplastic coupling isn’t considered in establishing the mechanics model of the milled mixture of sugar cane. In order to describe elastoplastic coupling and yield characteristics, based on uniaxial confined compression tests and direct shear tests, the evolution of the elastic parameters of the solid fiber with void ratio and the plastic deformation, and the plastic mechanics behavior of the solid fiber is described by Modified Drucker-Prager Cap model, the expressions of the parameters of the plastic mechanics model are derived and the corresponding parameters are calculated; and the elastoplastic coupling mechanics model is established. The evolution of the parameters of the model is achieved by the custom subroutine written in Fortran, the numerical simulation of the elastoplastic coupling model is realized by ABAQUS. The results of uniaxial confined compression tests and finite element simulation show that the elastoplastic coupling mechanics model is more accurate than the non-elastoplastic coupling mechanics model to predict the axial pressure. The results provide an important reference for the analysis and understanding of the milling process of sugar cane and the establishment of an accurate mathematical model of the milled mixture of sugar cane.

Coupled Modeling of Sedimentary Basin and Geomechanics: A Modified Drucker–Prager Cap Model to Describe Rock Compaction in Tectonic Context

The aim of basin modeling is to characterise fluids and rocks in a basin considering its history and data partly describing its present state. In usual basin simulators, only a simplified description of geomechanics based on the hypothesis of oedometric strain is used. To both enhance the modeling of basin history and to characterise actual in situ stresses, the effect of stress redistribution, horizontal stresses, and strain variations during basin history should be considered. To address this point, a coupled basin-geomechanics framework based on a new constitutive law is proposed in this paper using the prototype simulator $$\mathrm{A}^{2}$$ . This framework has been built to provide relevant results for various kinds of basin cases including tectonic loading. A finite strain poromechanical approach is considered along with an modified Drucker–Prager Cap model to describe rock compaction under natural sedimentation, erosion, and tectonics. The constitutive model can be seen as a tensorial extension of the compaction models of Athy or Schneider as it allows to recover the same behaviour in oedometric context. Simple test cases are modeled considering typical sand or shale properties, emphasizing the effect of tectonic loading on the present-day pore pressures and in situ stresses. It appears that even relatively moderate tectonic loading ( $$5\%$$ of horizontal strain) can lead to overpressures of several hundreds of bars and to a complete change in in situ stress regime for deeply buried layers (above a depth of 2000 m).

Investigation on compaction behaviors of Ag35Cu32Zn33 mixed metal powders under cold die compaction

Series experiments were performed to investigate the cold die compaction behaviors of Ag35Cu32Zn33 mixed metal powders to manufacture thin sheet of cadmium-free silver based filler metal, and the effect of friction on powder compaction behaviors were analyzed using the finite element method (FEM). The density dependent modified Drucker-Prager Cap model was established to characterize the mixed metal powders compaction behaviors. The die wall friction coefficients under different lubricated conditions were experimentally determined and were modeled by Coulomb friction model in simulations. The established constitutive model and material parameters were validated by experiments and finite element simulations of powder compaction process in ABAQUS/Standard with a user subroutine USDFLD. The compaction mechanisms of Ag35Cu32Zn33 mixed metal powders were analyzed with Gerdemann-Jablonski compaction equation, and the nonlinear compaction equation considering the friction effect was established and validated to characterize the powder compaction mechanisms under different die wall friction. With the friction coefficient increased, the relative fractional contributions of the particle rearrangement mechanisms increased and the powder deformation mechanisms contributions decreased. Finally, the influences of die wall friction on powder compaction behaviors were discussed. The results shown that the die wall friction significantly influence the powder flow behaviors and residual elasticity, which induce inhomogeneous relative density and stress distribution, cracks and capping in powder compact.

Identification of the modified Drucker–Prager yield condition and modeling of compaction of the plasticized titanium feedstock

A computer modeling procedure of densification of the noncompact titanium feedstock is considered. The modified Drucker–Prager cap model is used to describe the rheological flow of the deformed mass. It is shown that, when identifying the accepted model with the accuracy acceptable for engineering calculations, it is reasonable to use the auxiliary curve based on the Bernoulli lemniscate, which makes it possible to reduce the number of experiments necessary to construct the piecewise-smooth Drucker–Prager yield curve. The plastic deformation of the representative volume cell of sieving the titanium sponge in various deformation modes is investigated. To improve the formability of the noncompact titanium-containing feedstock, the plasticizing effect associated with an increase in the amount of the plastic β-phase upon hydrogen alloying is used. It is revealed based on theoretical and experimental investigations that hydrogen alloying makes it possible to form a denser billet at invariable temperature and compaction force compared with the traditional densification technology of titanium sponge. It is established that the distribution uniformity of relative density over the axial billet section increases with the additional hydrogen alloying. The satisfactory convergence of results of computer modeling and an experimental investigation into the compaction of the titanium sponge in a closed die is shown.

Experiment and finite element analysis of compaction densification mechanism of Ag-Cu-Sn-In mixed metal powder

Experiments and finite element analysis were performed to investigate the compaction densification mechanism of Ag57.6-Cu22.4-Sn10-In10 mixed metal powder. Firstly, the friction coefficient for lubricated and unlubricated die between the powder and die wall were determined experimentally. It is observed that the friction coefficient varies with the top punch pressure, and μ increases with the top punch pressure at low pressure regime (≤100MPa), while maintain constants at high pressure regime (>100MPa). The difference between the friction coefficients of lubricated/unlubricated die indicates that the lubrication (zinc stearate alcohol solution) could reduce the die wall friction greatly. Furthermore, the relative density distribution in the compacts for lubricated and unlubricated die compaction were characterized and mapped experimentally using the Vickers hardness method. Globally, the agreement between simulation calculation and experimental measurement of relative density distribution is reasonably good, indicating that the model parameters give an adequate description of the compaction behavior for the mixed metal powder. Moreover, a nonlinear compaction equation containing the die wall friction effect was proposed and established as: ln1−D1−D0=kP1−k1μ+b1−k2μP, k1 and k2 are correction coefficients related to the powder property. It is shown that the established nonlinear compaction equation could represent the relative density-compaction pressure relationship for the mixed metal powder under different die wall friction. At last, the influence mechanisms of die wall friction on the compaction behavior of mixed metal powder were detailedly analyzed in simulations using the modified Drucker-Prager Cap model. The results show that the friction induced inhomogeneous stress distribution and powder flows in compact during compaction, which caused density gradient and the occurrence of cracks and capping during ejection.

Physical interpretations for cap parameters of the modified Drucker-Prager cap model in relation to the deviator stress curve of a particulate compact in conventional triaxial testing

The physical meanings of the cap aspect ratio (R) and transition surface parameter (α) of the modified Drucker-Prager cap (MDPC) model have been uncovered in relation to the deviator stress curves of a particulate material in conventional triaxial testing by simulating the curves using varying R and α based on finite element analysis. R controls the rate of the stress rise with the increase of the strain; the smaller the R, the faster the rise of the deviator stress. This phenomenon occurs because, in the p-q plane (p is the mean stress and q is the Mises equivalent stress), the cap with a smaller R needs to move a shorter distance on the p axis to maintain the current stress state: a smaller volumetric plastic strain is required according to the hardening law. R does not influence the maximum value of the deviator stress curve. As for the influence of α, it artificially lowers the true failure surface by an amount that is proportional to α so that a fictitious ultimate failure state is achieved. Therefore, it is desirable to set α as small as possible unless the numerical analysis using the MDPC model does not produce a converged solution. An analytical expression to calculate the maximum deviator stress that can be predicted by the MDPC model is provided in terms of α.

A modified Drucker–Prager Cap model for finite element simulation of doped tungsten powder compaction

This paper presents a density dependent modified Drucker–Prager Cap (DPC) model for the compaction behavior of potassium doped tungsten powder. The experimental calibration of the model is done using an instrumented cylindrical die in powder compaction tests. The die is designed for direct determination of loading and unloading forces of powder compact inside the die in compaction and transverse directions. The cap surface parameters are characterized by fitting stress and strain curves acquired during loading to different green density values. The material parameters for the DPC failure surface are obtained by subsequent diametrical and axial compression tests. Additionally the contact pressure dependent wall friction coefficient is determined. The material model is implemented in the ABAQUS finite element code by user subroutines. The calculated compaction forces for a double-action compaction process are in excellent agreement with experimental data. The results show, that the proposed material model for doped tungsten can be used in finite element simulations of the compaction process.

A simulation-based determination of cap parameters of the modified Drucker–Prager cap model by considering specimen barreling during conventional triaxial testing

Two types of cap parameters were determined by fitting the experimentally obtained triaxial deviator stress curves found in the literature using the modified Drucker–Prager cap model. The two types of parameters were extracted using the specimen model that deforms uniformly or non-uniformly (barreling). The parameter set extracted from the non-uniformly deforming specimen model reasonably predicted the experimental deviator stress curves obtained from the barreling specimen. However, when the parameter set extracted from the uniformly deforming specimen model was used to simulate the experimental curves from the barreling specimen, the prediction was inferior to the set from the non-uniformly deforming model. Considering versatile types and properties of particulate materials (e.g., ceramic, metal, pharmaceutical powders and soils), we suggest to verify the prediction capabilities of the parameter sets from each specimen model. The stress path of the barreling specimen during the conventional triaxial test was fairly complex depending on the position in the specimen. Pure hydrostatic loading was not achieved even during the hydrostatic loading stage. In the shear loading stage, the position-averaged stress path showed a lower slope in the meridional plane than that of the uniformly deforming specimen with a slope of 3.

OS1727 鉄系圧粉体の材料モデルパラメータの推定

In order to estimate parameters of the simulation model for iron green bodies of press-ready premix consisting of Distaloy AE, 0.5 % graphite and 0.6 % Kenolube, we carried out diametrical compression tests of the iron green bodies. We prepared the iron body with densities of 6.9 g/cm^3, 7.2 g/cm^3, 7.4 g/cm^3. The parameters of failure envelope in the modified Drucker-Prager Cap model were determined using the results of uniaxial compression tests in our previous study and diametrical compression tests.

Numerical simulation of residual stress field in green power metallurgy compacts by modified Drucker–Prager Cap model

Compaction process simulation and residual stress prediction of green PM compact were carried out with elasto-plastic 3D FEA based on the modified Drucker–Prager Cap model in Abaqus. The model parameters of the investigated powder Distaloy AE were determined as functions of relative density through typical mechanical property tests of powder. The model was implemented as a user subroutine USDFLD. Single sided compaction of a d20 mm×5 mm disk green compact of Distaloy AE was simulated, and the residual stress of the disk after ejection was predicted with FEA. The FEA results of the compaction process and the residual stress of the disk show good agreement with compaction experiments and X-ray diffraction measurements, which validates the model and its parameters. The results indicate that the compressive residual stresses exist mainly in a thin layer on the side surface, but the residual stresses are very small on the top and bottom surfaces.

Modelling of the break force of tablets under diametrical compression

A numerical method to predict the break force of curved faced tablets is proposed. The constitutive model and the failure criteria necessary to obtain predictions consistent with experimental data are identified. A modified Drucker-Prager cap model together with a maximum principal stress based failure criteria was found sufficient to predict the break force of tablets under diametrical compression loading. The conditions for the validity of the method were identified with reference to practical tablet shapes and failure patterns. Under these conditions the numerical procedures can be used as a practical tool to predict tablet breakage as an alternative to the empirical methods currently used in pharmaceutical product design and process development.

Calibration of a Density-Dependent Modified Drucker-Prager Cap Model for AZO Powder

AZO (Aluminum doped Zinc Oxide) is widely used to produce transparent conductive coatings for liquid crystal displays, flat panel displays, plasma displays, touch panels, and electronic ink applications. The densification behavior of AZO powder is a critical factor related to the design of the compaction process. In this study, the densification behavior of AZO powder during cold compaction has been investigated in order to calibrate the modified Drucker-Prager Cap (DPC) model for FE simulations. A compaction test with a cylindrical die was carried out, and two failure tests were performed: the diameteral compression test and the uniaxial compression test. AZO compacts with various densities from the compact tests were used as specimens for the failure tests. Based on the experimental results, the parameters of the modified DPC model were determined through simple manipulations.