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

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 (p: hydrostatic stress and q: Mises equivalent
stress). In this plan, the failure surface of the model consists of two
principal parts: the failure line (F_s) and the cap
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 p_a, cap eccentricity
R and hydrostatic yield pressures p_b) are
calculated from d and b.
Thus the variation of R, p_a and p_b
is due to the variation of d and 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 d and 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
±0.53s_d
and s_u
±=s_u
± 0.53s_u).
The corresponding values of b and
d were then calculated.

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-q plan. The equation
of the yield surface (yield function F) is defined as:

Where a is the size of the yield surface
and M is a material constant.

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 E and 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-,
n+ (with variations of ±20%
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). E varies from 0.14
to 1.9 GPa at -20% and from 0.21 to 2.8 GPa at +20%. n is
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 E has no influence
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 E has no effect on these output variables at the
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.