# (113d) Sensitivity of Plastic and Elastic Parameters during the Numerical Simulation of Pharmaceutical Die Compaction Process with Drucker-Prager/Cap Model

#### World Congress on Particle Technology

#### 2018

#### 8th World Congress on Particle Technology

#### Handling & Processing of Granular Systems

#### Finite Element Modeling of Granular Materials

#### Wednesday, April 25, 2018 - 4:36pm to 4:58pm

Finite element simulation with Drucker-Prager/Cap (DPC)

model is commonly used to conduct numerical modeling of the compression of

granular material. In the pharmaceutical field, this model is increasingly used

for the numerical simulation of the process of die compaction.

The compression curve, obtained from die compaction

process, consists of three parts: the loading part (compression phase), the

unloading part (decompression phase) and the ejection phase. Commonly the axial

stress (applied stress) as a function of the tablet thickness and the radial

transmission (radial stress versus axial stress) curves are studied. In

addition of the compression curves, the numerical modeling makes it possible to

analyze the spatial distribution of the density and the stresses.

To perform numerical simulation with DPC model, the model

parameters must be determined. Usually die compaction test is used to make

compacts and, during the compression, data are recorded. The compacts are then

treated and submitted to at least two failure tests, and new data are collected.

The data obtained from die compaction and failure tests make it possible to

determine the DPC parameters.

Depending on the method used, the instrumentation of the

devices or the accuracy of the measures, the data obtained from the

experimental tests for the characterization of the model parameters may vary

from one study to another. For granular materials like pharmaceutical powders,

this variation can be important and the consequences on the final result must

be studied. In this work, the influence of the plastic and elastic parameters

on the numerical results during the compression and the decompression phases will

be discussed.

** Influence of plastic parameters of DPC
model**

The DPC model is commonly presented in the ** p**-

*q*plan (

**: hydrostatic stress and**

*p***: Mises equivalent**

*q*stress). In this plan, the failure surface of the model consists of two

principal parts: the failure line (

**) and the cap**

*F*_{s}surface (

**).**

*F*_{c}For the failure line, the parameters to characterize are

the powder cohesion ** d** and the friction angle

**.**

*b*The cap parameters (hydrostatic pressure

**, cap eccentricity**

*p*_{a}**and hydrostatic yield pressures**

*R***) are**

*p*_{b}calculated from

**and**

*d***.**

*b*Thus the variation of

**R**,

**and**

*p*_{a}

*p*_{b}is due to the variation of

**and**

*d***.**

*b*The parameters of the failure line are usually characterized by using diametral

and uniaxial compression tests. Eq.1 can be expressed as a function of failure

strengths obtained from diametral (

*s***)**

_{d}and uniaxial (

*s***)**

_{u}compression tests and

**and**

*d*

*b*can be calculated as:

All these parameters were characterized by experimental

tests and represent the reference values in this work. Eq.3 shows that potential

variations of ** b** and

*d*are necessarily due to variations of

*s*

_{u}and

*s***.**

_{d}Thus, in the proposed work, variations of ±53%

were applied to

*s*

_{u}and

*s***.**

_{d}These variations were applied in order to cover a broad range of the friction

angle values of microcrystalline cellulose (MCC) found in the literature. Five

cases were studied: the reference case (

*s***,**

_{d}

*s***)**

_{u}and four other cases: (

*s*

_{d}+,

*s*

_{u}+), (

*s*

_{d}-,

*s*

_{u}+), (

*s*

_{d}+,

*s*

_{u}-) and (

*s***-,**

_{d }

*s*

_{u}-) with (

*s*

_{d}±=

*s*

_{d}±0.53

*s*

_{d}and

*s*

_{u}±=

*s*

_{u}± 0.53

*s***).**

_{u}The corresponding values of

**and**

*b***were then calculated.**

*d*Numerical simulations were conducted with variations of

the friction angle and the powder cohesion for the five cases cited above. The

plot of the axial stress as a function of the tablet thickness shows that the

curves obtained from the five cases are identical. The curves of radial stress

versus axial stress are also the same. Thus, during the compression and the decompression

phases, variations of the friction angle and/or the cohesion of the powder have

no influence on the evolution of these output variables.

Since the variation of the failure line has no influence

on the results during the compression and decompression phases and the

characterization of its parameters requires additional experimental tests, an

alternative model like the modified Cam-Clay (MC-C) model can be used to

simulate the compaction. The MC-C model is simple to use and requires less

experimental tests for the parameter characterization. The yield surface has the

shape of half an ellipse in the ** p**-

**plan. The equation**

*q*of the yield surface (yield function

**) is defined as:**

*F*Where ** a** is the size of the yield surface

and

**is a material constant.**

*M**Comparison
between DPC and MC-C models*

DPC and MC-C models were used to simulate the powder

behavior during die compaction process. The curves of axial stress *p _{ax}*

versus tablet thickness obtained from MC-C and DPC models are superimposed. The

radial transmission obtained from MC-C model is identical to that obtained from

DPC model. These comparisons show that the simulation with the two models gives

the same result.

The axial stress distributions are the same for the two

models at the maximum of compression and at the end of decompression. The same

result was found for the relative density distribution. It is clear that the

MC-C model is able to predict the powder behavior during the compression as

well as the DPC model.

**Influence of elastic parameters of DPC
model**

Variations of ±20% on the values of Young’s modulus *E*

and Poisson’s ratio ** n** obtained

from experiments were applied. This makes it possible to cover a broad area of the

values of

**and**

*E*

*n*found in the literature for the numerical modeling of MCC powder with DPC model.

Nine configurations were studied: the configuration with

reference values of ** E** and

*n*(reference case), the cases

**-,**

*E***+,**

*E***-,**

*n***+ (with variations of ±20%**

*n*on only one of the two parameters) and the cases

**-**

*E***-,**

*n***+**

*E***+,**

*n***-**

*E***+**

*n*and

**+**

*E***-**

*n*(with variations of ±20% on both parameters).

**varies from 0.14**

*E*to 1.9 GPa at -20% and from 0.21 to 2.8 GPa at +20%.

**is**

*n*between 0.18 and 0.32 at -20% and between 0.26 and 0.48 at +20%. In comparison

to the elastic properties found in the literature, these variations are

realistic. The numerical curves (axial stress versus tablet thickness and

radial stress versus axial stress) for all the configurations (Ref,

**-,**

*E***+,**

*E***-,**

*n***+,**

*n***-**

*E***-,**

*n***+**

*E***+,**

*n***-**

*E***+**

*n*and

**+**

*E***-)**

*n*were analyzed.

*Influence
of Young’s modulus*

The variation of Young’s modulus ** E** (Poisson’s

ratio was kept constant) leads to a change of the axial stress versus tablet

thickness curve. The difference appears at the beginning of the powder

hardening and remains until the end of the compression phase. During the

decompression phase, this difference decreases and disappears at the end of the

decompression. As a consequence, the change of

**has no influence**

*E*on the final tablet thickness.

The variation of Young’s modulus ** E** has no

influence on the radial transmission curve. The influence of

*E*was also assessed at a local scale by comparing the distributions of the

relative density, the axial and radial stresses at the maximum of compression

and at the end of the decompression. The distributions are always identical

which means that

**has no effect on these output variables at the**

*E*local scale.

*Influence
of Poisson’s ratio*

The influence of Poisson’s ratio *n*

gradually appears during the powder hardening (during plastic deformation) and

is maximal at the end of the compression phase. For a same compression

pressure, the tablet thickness is lower when a small Poisson’s ratio is used.

During the decompression phase the effect of Poisson’s ratio slightly decreases

but the final tablet thicknesses remains clearly different.

Concerning the radial transmission, at low pressure, the

variation of ** n** does not lead to a

change of the die wall pressure (radial stress). When the applied pressure is high

enough (up to 120 MPa), the die wall pressure increases with

**.**

*n*At the end of the decompression, a small Poisson’s ratio gives a high residual

die wall pressures (RDP) and the increase of

*n*leads to a decrease of the RDP.

The distribution of the relative density does not change

when Poisson’s ratio varies. The same result was found for the distribution of

the axial stress at the maximum of compression. For the distribution of the

radial stress at the maximum of compression, a slight difference is observed. The

stress distribution at the end of the decompression phase is completely dependent

on Poisson’s ratio. With small Poisson’s ratio, the residual stress is high and

it is low when ** n** is

high.

This study shows that the influence of Poisson’s ratio ** n **is

more important than that of Young’s modulus. The influence of

*n*is shown to be less important on the MDP than on the RDP. As a consequence, an

underestimation of Poisson’s ratio may not show a difference in the MDP but may

strongly overestimate the RDP. In addition the occurrence of capping is correlated

to the RDP. Thus, when studying capping problem with FEM modeling during die

compaction, Poisson’s ratio is an important parameter and must be characterized

with accuracy.