Drucker Prager Cap Research Articles

Comparative study between Drucker-Prager/Cap and modified Cam-Clay models for the numerical simulation of die compaction of pharmaceutical powders

Numerical simulation with finite element method is commonly used to model the process of die compaction. In pharmaceutical field, the Drucker-Prager/Cap (DPC) model is by far the most used to study the mechanical behavior of the powder. Another classical model but less used model, the modified Cam-Clay (MC-C) model, is introduced in this study. The main advantage is that the model parameters are easier to determine for MC-C model than for DPC model.The yield surface of the MC-C model is an elliptical curve. This model was compared to DPC model in order to assess its ability to describe the powder behavior in compression. Experimental tests were done to characterize the parameters of both models and the simulations were performed.The global behavior of the powder in compression was assessed from the equations of the models and the plots of compression curves. The spatial distributions of the relative density and axial stress were studied to compare the local behavior of the compact (stress concentration, areas of low/high density). The results obtained from the two models were nearly identical in the case of flat-faced punches but also when using concave-face punches. This means that the MC-C model is able to well describe the die compaction of pharmaceutical powder.

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.

Effect of sand dilation on distortions and pattern allowances during steel sand casting

Mechanical interactions between the sand mold and casting have a great impact on pattern allowances. In this study, the effect of core expansion on distortions during steel casting is investigated. A hollow steel cylinder is cast using silica and zircon sand cores. The evolution of the cylinder’s inner diameter is measured in situ using LVDTs. During solidification, core expansion is found to distort the inner diameter into a barrel-shaped profile, generating the largest expansion at the mid-height. The experiments are simulated using a sequential thermo-mechanical coupling. In the stress analysis, the steel and sand are modeled using an elasto-visco-plastic constitutive law and the Drucker–Prager Cap model, respectively. The simulations reveal that sand dilation due to shear stresses, as opposed to thermal expansion, accounts for the majority of the increase in the inner diameter. The measured and predicted pattern allowances are found to be in excellent agreement.

The Application of Numerical Analysis in Determining the State of the Rock Mass Around Directional Wells

Drilling directional and horizontal wellbores plays an important role in the exploitation of hydrocarbons. In the case of long operated and unconventional fields the importance of this type of drilling is fundamental. The horizontal and directional drilling technologies are also applicable in other fields. Directional and horizontal holes are among other various types of drilling under buildings, microtunnels, injection and storage holes. This technology is also used in obtaining coal-bed methane and in underground coal gasification methods.Compared to holes with vertical axes, directional drilling is a much more complex issue. The curvature of the wellbore's axis, passing through the horizontally stratified rock mass, causes the conditions around its cross section to be strongly anisotropic. Not infrequently, the same rock material of each layer exhibits anisotropic properties. Ensuring the stability of the well is very difficult under such conditions.The proper determination of the state of stress, strain and displacement of the rock mass around the hole is critical in the design of directional holes. Numerical analysis tools which take into account the structure of the rock mass and the mechanical properties of rocks along the well trajectory are used for this purpose.The article presents numerical simulations of changes of the rock mass state around a directional well. The calculations were conducted with various material models of rock: the linear-elastic model, the linear-elastic with regard to pore fluid flow model, the elastic – plastic model, the Drucker-Prager cap model with pore fluid flow, the concrete damage plasticity model (CDP), and the cracked material model. The impact of primary pressure, lamination and anisotropy of the rock mass are also included in the calculation. Based on the results of the calculations we analyze the impact of the behavior of rocks on the wellbore's stability.

Prediction of distortions and pattern allowances during sand casting of a steel bracket

Mechanical interactions between the casting and mould generate unwanted distortions and lead to dimensional inaccuracies. In this study, the effects of mould expansion and mould restraint are investigated through sand casting experiments involving a U-shaped steel bracket. Distortions are quantified by in situ measurements of the evolution of the gap opening between the bracket legs. Mould expansion is observed immediately after filling. Outer mould restraint prevents distortions in the bracket legs until the time of mould fracture, after which the legs are pushed outward. The experiments are simulated using a sequential thermo-mechanical coupling. The steel and bonded sand are modelled using previously calibrated elasto-visco-plastic and Drucker–Prager Cap constitutive laws, respectively. Excellent agreement between measured and predicted pattern allowances (PA) is obtained. Distortions are greatly under-predicted unless mould fracture is considered. Variations in the packing density of the moulds are also shown to have an impact on PA.

Smooth Yield Surface Constitutive Modeling for Granular Materials

In this paper, the authors present an internal state variable (ISV) cap plasticity model to provide a physical representation of inelastic mechanical behaviors of granular materials under pressure and shear conditions. The formulation is dependent on several factors: nonlinear elasticity, yield limit, stress invariants, plastic flow, and ISV hardening laws to represent various mechanical states. Constitutive equations are established based on a modified Drucker–Prager cap plasticity model to describe the mechanical densification process. To avoid potential numerical difficulties, a transition yield surface function is introduced to smooth the intersection between the failure and cap surfaces for different shapes and octahedral profiles of the shear failure yield surface. The ISV model for the test case of a linear-shaped shear failure surface with Mises octahedral profile is implemented into a finite element code. Numerical simulations using a steel metal powder are presented to demonstrate the capabilities of the ISV cap plasticity model to represent densification of a steel powder during compaction. The formulation is general enough to also apply to other powder metals and geomaterials.

Numerical analysis of the dynamic interaction between a non-pneumatic mechanical elastic wheel and soil containing an obstacle

An innovative non-pneumatic tyre called the mechanical elastic wheel is introduced; significant challenges exist in the prediction of the dynamic interaction between this mechanical elastic wheel and soil containing an obstacle owing to its highly non-linear properties. To explore the mechanical properties of the mechanical elastic wheel and the soil, the finite element method is used, and a non-linear three-dimensional finite element wheel–soil interaction model is also established. Hyperelastic incompressible rubber, which is one of the main materials of the mechanical elastic wheel, is analysed using the Mooney–Rivlin model. The modified Drucker–Prager cap plasticity constitutive law is utilized to describe the behaviour of the soil, and the obstacle is represented as an elastic body. Simulations with different rotational speeds of the mechanical elastic wheel were conducted. The stress distribution and the displacement of the mechanical elastic wheel and the soil were obtained, and the effects of different rotational speeds on the displacement, the velocity and the acceleration of the hub centre are presented and discussed in detail. These results can provide useful information for optimization of the mechanical elastic wheel.

A density-dependent modified Drucker-Prager Cap model for die compaction of Ag57.6-Cu22.4-Sn10-In10 mixed metal powders

Due to the bad plastic workability, silver based brazing filler metals with high proportions of additional elements such as Zn, Sn, In, Ga, are very brittle and difficult to be made into thin sheets using conventional manufacturing processes. A novel process method that combine powder compaction and sintering, can fabricate thin sheets of the fragile silver based filler metals. In this work, the densification mechanism of Ag57.6-Cu22.4-Sn10-In10 (wt.%) mixed metal powders was investigated, and a modified density-dependent Drucker-Prager Cap (DPC) model was introduced to describe its compaction behavior based on the equivalent density method. The powder compact was considered as a continuum, and a linear elasticity law as a function of relative density was then used to express the elastic behavior of the powders. An instrumented die with force transducers has been designed to determine the material parameters of the modified DPC model. The elastic and plastic material parameters such as E, υ, β, d, pa, R and pb were then determined experimentally. The friction coefficient between the powders and the die wall with a lubricated die was determined based on the Janssen–Walker theory in the instrumented die experiments. Furthermore, the modified DPC model with the linear elasticity law was validated using finite element simulation method in ABAQUS with a user subroutine (USDFLD). A good agreement between the simulations and experimental results was obtained, indicating the feasibility and applicability of the introduced modified models to describe the die compaction behavior of the mixed metal powders.

Numerical simulation tool for paving block structures assessed by means of full-scale accelerated pavement tests

ABSTRACTBlock pavements are an attractive alternative to asphalt and concrete pavements, especially in communal areas. Architects and urban planners would like to take advantage of various shapes, colours and textures of paving blocks in order to achieve a higher quality of urban space if the performance of block pavements could be better predicted to avoid large horizontal displacements of chipped stone corners and rutting. Unfortunately, the computational performance prediction of paving block structures is more complex than that for flexible pavements with homogeneous surface layers of asphalt concrete. The influence of the large number of vertical joints between paving blocks on the overall mechanical performance has not been considered sufficiently within computational tools yet. The proposed numerical simulation tool is able to take into account the complex mechanical behaviour of sand-filled joints as well as the non-linear mechanical behaviour of the underlying base courses. Joints are modelled using a Mohr–Coulomb-type friction model with the normal stresses non-linearly related with joint opening. Three different experimental set-ups were developed for the identification of the model parameters. The base behaviour was modelled using the Drucker–Prager cap model. The paper shows that the proposed tool predicts reasonable deformations and stresses in block pavements. The results of the simulations were compared with measured stresses from the full-scale accelerated pavement testing and a good agreement was observed.

Modeling the Formation of Debossed Features on a Pharmaceutical Tablet

The objective of this work was to develop, validate, and implement a modeling methodology for predicting the shape and relative density fields in the vicinity of debossed features on a tablet surface. The resulting model was used to investigate the influence of debossed feature stroke angle and degree of pre-pick, which is expressed as a percentage of the stroke depth, as well as the influence of formulation lubricant on the aforementioned debossed feature parameters. An experimental procedure for measuring formulation (modified) Drucker–Prager Cap parameters is described. These parameters are used in a finite element method simulation that models the formation of a debossed surface feature on a tablet. Techniques for validating the simulation and post-processing the results are also described. The stroke angle and degree of pre-pick significantly influence the debossed feature dimensions, with larger degrees of pre-pick and stroke angles giving debossed features that more closely match the target (embossment) values. Lubrication plays a much weaker role, but did improve the fidelity of the debossed feature slightly. The differences between the debossed and target feature dimensions are due to elastic spring back of the material. The tablet relative density is smallest at the shoulders of the debossed feature and largest at the base of the valley. Although the relative density fields show no obvious trends with stroke angle, the fields are clearly more uniform as the degree of pre-pick increases. The addition of lubricant to the formulation also improves the relative density field uniformity for larger degrees of pre-pick. To improve feature fidelity and decrease the likelihood of damage, larger pre-picks, larger stroke angles, and the addition of a formulation lubricant should be used.

Effects of the granule composition on the compaction behavior of deformable dry granules

Calibration of the Drucker Prager Cap (DPC) model parameters provides a means for a deeper understanding of the impact of granule composition on the compaction properties of dry granules independent of their solid fraction. In this study, monodisperse granules of mixtures of microcrystalline cellulose and mannitol (0%, 25%, 50%, 75% and 100% mannitol) prepared as small cylindrical compacts with well-defined size, shape and solid fraction (0.58) were used as model dry granules. DPC parameters–namely, cohesion, internal friction angle, cap eccentricity, and hydrostatic yield strength of materials–were determined from the diametrical and uniaxial compression, and in-die compaction tests. Elastic properties such as Young's modulus and Poisson's ratio were also determined from the in-die compaction test. Higher level of mannitol in granules required a lower compression pressure to obtain a low solid fraction tablet but higher compression pressure to obtain a high solid fraction tablets. Properties such as cohesion and diametrical tensile strength go through a maximum as the mannitol level increases in the binary granules, and clearly do not follow a simple linear mixing rule. At an industrially-relevant tablet solid fraction of 0.88, granules with 75% mannitol exhibited the highest cohesion, and produced the strongest tablet. Other properties either approximately follow the linear mixing rule (e.g., hydrostatic yield strength, Young's modulus and Poisson's ratio) where limited interactions between the constituents are present, or not sensitive to the composition (e.g., internal angle of friction). In general, the compaction behavior of granules of a multicomponent system may not be precisely estimated from the properties of individual components, simply by using the linear mixing rule.

Compaction mechanics of plastically deformable dry granules

To improve the understanding of how dry granulation and in particular, granule solid fraction (SF) impact the compaction behavior of plastically deformable microcrystalline cellulose (MCC), in this study, the Drucker Prager Cap (DPC) model parameters were calibrated using monodisperse MCC dry granules as model granules. Dry granules were produced as directly compressed small cylindrical compacts of MCC with SF in the range of 0.40 to 0.70 which were monodisperse in both size and SF. Virgin MCC powder and granules were compressed into tablets with SF in the range of 0.70 to 0.90. The DPC parameters (cohesion, internal friction angle, cap eccentricity, and hydrostatic yield stress), Young's modulus and Poisson's ratio were experimentally determined from diametrical and uniaxial compression, and in-die compaction tests. Results showed that calibration of the shear failure surface only may be adequate for MCC granules when the DPC model is completely calibrated for virgin MCC. Increasing granule SF significantly decreased the cohesion only. All other parameters were impacted by the tablet SF only. In the 2D yield surface, only the shear failure surface expanded as the granule SF increased. MCC of any granulation status requires the same in-die compaction stress state for densification to a given tablet solid fraction.

Effect of roll compactor sealing system designs: A finite element analysis

In the pharmaceutical industry, the roll compaction is part of the dry granulation process, densifying fine powders into ribbons that will be later milled to produce granules with good flowability for subsequent die compaction process. Roll compactors are constructed with a sealing system, limiting the loss of powder from the sides. However, the sealing system may result in unwanted non-uniformity of the ribbon's properties. In this work, a 3D Finite Elements Method (FEM) modeling is used to analyze the roll compaction process and the effect of sealing system designs on the compacted ribbon's density distribution. A density dependent Drucker–Prager Cap (DPC) constitutive model for microcrystalline cellulose (Avicel PH-101) was calibrated and implemented in Abaqus/Explicit. Two different FEM models were investigated, one with a fixed side sealing called cheek plates and another where the side sealing is integrated with the bottom roll called rimmed-roll. Both numerical and experimental results clearly show the non-uniform roll pressure and density distribution for the cheek plates assembly, whereas the rimmed-roll shows an overall more uniformly distributed resultant pressure and density distribution. These results demonstrate the capability of FEM modeling to provide insight and help achieving a better understanding of the roll compaction process.

Finite strain elastoplasticity considering the Eshelby stress for materials undergoing plastic volume change

In consideration of materials capable of undergoing significant plastic changes in volume, an alternative finite strain hyper-elastoplastic constitutive framework is proposed in terms of the Eshelby stress. Taking a phenomenological point of view, a thermodynamically-consistent approach to developing the constitutive equations is presented and discussed. Various Eshelby-like stresses are defined and shown to be energy-conjugate to the plastic velocity gradient, and a general framework is formulated in the stress-free/plastically-deformed intermediate configuration associated with the multiplicative split of the deformation gradient, as well as the current configuration. A novel Eshelby-like stress measure is proposed, which is scaled by the elastic Jacobian, and is shown to be energy-conjugate to the plastic velocity gradient in the spatial representation. Modified Cam–Clay and Drucker–Prager cap plasticity constitutive equations are introduced, and large strain isotropic compression simulations are performed and compared with experimental measurements. The model results are compared with standard approaches formulated in terms of the Mandel and Kirchhoff stresses, which are shown to require the assumption of isochoric plasticity to satisfy the Clausius Planck inequality (Mandel) and preserve that the intermediate configuration remains stress-free (Kirchhoff). The simulations show that both the material and spatial Eshelby-like stress measures presented here produce the same mean Cauchy stress results; whereas, standard formulations, which make use of isochoric plasticity assumptions, diverge from each other at significant plastic volume strains. Standard formulations are further shown to violate the second law of thermodynamics under certain loading conditions. Calibration of model parameters to high pressure isotropic compression of Boulder clay is used to compare the various models.

Materialni parametri za numerično simulacijo postopka stiskanja sintranih dvovišinskih zobnikov

This paper presents the initial material parameters for the Ecka Alumix 231 aluminum-powdered metal required for a successful numerical simulation of a compaction process for sintered components with a proper software package. Experimental work was used to obtain Drucker-Prager-cap (DPC) model material parameters with the help of a Brazilian disc test and a uniaxial compression test for the linear part of the DPC model in the equivalent pressure stress/deviatoric stress (p-q) plane. The specimens used for this test were cylindrical probes (greens) that were compacted with a mechanical press and then compressed to the failure point in the axial and radial directions.

Investigating the effect of tablet thickness and punch curvature on density distribution using finite elements method

Finite elements method was used to study the influence of tablet thickness and punch curvature on the density distribution inside convex faced (CF) tablets. The modeling of the process was conducted on 2 pharmaceutical excipients (anhydrous calcium phosphate and microcrystalline cellulose) by using Drucker–Prager Cap model in Abaqus® software. The parameters of the model were obtained from experimental tests.Several punch shapes based on industrial standards were used. A flat-faced (FF) punch and 3 convex faced (CF) punches (8R11, 8R8 and 8R6) with a diameter of 8mm were chosen. Different tablet thicknesses were studied at a constant compression force.The simulation of the compaction of CF tablets with increasing thicknesses showed an important change on the density distribution inside the tablet. For smaller thicknesses, low density zones are located toward the center. The density is not uniform inside CF tablets and the center of the 2 faces appears with low density whereas the distribution inside FF tablets is almost independent of the tablet thickness.These results showed that FF and CF tablets, even obtained at the same compression force, do not have the same density at the center of the compact. As a consequence differences in tensile strength, as measured by diametral compression, are expected. This was confirmed by experimental tests.

The extrapolation of the Drucker–Prager/Cap material parameters to low and high relative densities

The present study explores new approaches to extract Drucker–Prager/Cap (DPC) constitutive model parameters at low and high densities of compacted powders for which it is not possible to get solid, undamaged samples for model calibration purposes. Extrapolations were carried out by invoking a number of physically plausible assumptions for high density conditions and the addition of the experimental shear cell testing procedure for low density extrapolations. The effects of these extrapolations on finite element model (FEM) results of both die compaction and roller compaction were examined. The sensitivity of the extrapolated DPC parameters on compaction model results was explored by performing parametric studies for both low and high density extrapolations. Examination of die compaction model results for low density showed little sensitivity to extrapolations; however, we are able to show that extrapolations of DPC parameters to low density may have a significant effect on roller compaction modeling results. High density die compaction FEM simulations reveal a significant effect on the way in which the DPC model parameters are extrapolated to high density. A method of extrapolating the DPC model parameters to high density is presented in this work. The work presented here demonstrates the significance of properly calibrating the DPC model at low and high densities and provides the necessary guidance for this purpose.

Dynamic spherical cavity expansion analysis of rate-dependent concrete material with scale effect

In this paper, dynamic spherical cavity expansion model is developed for rate-dependent concrete material described by modified Drucker-Prager Cap plasticity model incorporated with size effect. Penetration resistance components are computed accordingly with static resistance dependent on projectile diameter scale. Saturated region is introduced to investigate the response of concrete target under high impacting velocity. The validation of the proposed model is conducted by modelling a rigid projectile with full-scale and 1/10-scale diameter penetrating a semi-infinite concrete target. Depth of penetration numerically obtained is also compared with some widely recognized empirical penetration formulae, and better agreement is achieved by the proposed model for penetration tests with various striking velocities.

Effect of sand dilation on core expansion during steel casting

The thermo-mechanical behavior of the bonded sand used for molds and cores has a strong effect on dimensions of steel castings. Experiments are conducted in which a thick- walled hollow carbon steel cylinder is cast using a silica sand core. The temporal evolution of the inner diameter of the cylinder is measured in-situ during solidification and cooling by utilizing quartz rods connected to LVDTs (Linear Variable Differential Transformers). It is found that the inner diameter increases significantly during the initial stages of solidification when the steel offers little restraint to core expansion. Without accurately modeling this initial core expansion, the final cylinder dimensions at room temperature cannot be predicted. Preliminary simulations using the measured linear thermal expansion coefficient of the core considerably under-predict the measurements, which suggests that shear induced sand dilation also contributes to core expansion. The Drucker-Prager Cap model, which can predict dilative behavior, is used to simulate the mechanical behavior of the core. Utilizing this model in conjunction with an elasto-visco-plastic constitutive law for the steel, the stress simulations successfully predict the observed dimensional changes in the casting during solidification.