International Journal of Pharmaceutics 445 (2013) 99– 107
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International Journal of Pharmaceutics
jo ur nal homep a ge: www.elsev ier .com/ locate / i jpharm
odelling of the break force of tablets under diametrical compression
. Shang, I.C. Sinka ∗, J. Panepartment of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK
r t i c l e i n f o
rticle history:eceived 24 September 2012eceived in revised form 13 January 2013ccepted 16 January 2013vailable online 26 January 2013
a b s t r a c t
A numerical method to predict the break force of curved faced tablets is proposed. The constitutive modeland 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 criteriawas found sufficient to predict the break force of tablets under diametrical compression loading. Theconditions 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 topredict tablet breakage as an alternative to the empirical methods currently used in pharmaceuticalproduct design and process development.
Pharmaceutical tablets are made by compressing a powder mix-ure in a die. The tablets should be sufficiently strong to holdogether during post-compaction operations, packaging, storagend use. For a given powder formulation the strength of theablet depends on the density of the powder material, whichncreases with the compaction pressure used in manufacturing.fter administration the tablets should either disintegrate quicklyr release the drug in a controlled manner. These competing qual-ty requirements must be met while achieving high throughputates in manufacturing, e.g. the productivity of a typical rotaryablet press in manufacturing environment is of the order of.5–1 million tablets/h.
Tablet strength is therefore an important quality attribute. Therere many mechanical testing methods used to determine the tensiletrength of a material. The routine method used in the pharma-eutical industry is the diametrical compression test, whereby theablet is placed within rigid platens and loaded across the diameters illustrated in Fig. 1a. This is a standard test described in the cur-ent pharmacopoeias (United States Pharmacopoeia, 2011) and iserformed using standard testing equipment. The method is ofteneferred to in the pharmaceutical industry as “hardness testing”.
he basis of the method is Hertz’s contact theory (Timoshenko andoodier, 1970) which shows that when a thin disc made of a lin-ar elastic material is loaded across the diameter then a uniform
tensile state of stress develops through the centre of the specimen.For this reason the test method is also known as “indirect tensiletest” or “Brazilian test” (Carneiro and Barcellos, 1953; Akazawa,1953) which was used for measuring the tensile strength of rockfrom drill cores. The tensile strength is
�d = 2F
�Dt(1)
where �d is the tensile strength, F is the break force, D is the tabletdiameter and t is the tablet thickness.
Eq. (1) makes it possible to relate the brake force of a roundtablet with flat face to the tensile strength of the material, there-fore it is used widely in the design of pharmaceutical formulations,the development of manufacturing processes and quality control.For curved face geometry, however, the relationship between thebreak force in diametrical compression and the tensile strength ofthe material cannot be determined analytically using Eq. (1). Thisproblem was addressed by Pitt et al. (1988) who established anempirical relationship between the break force of convex shapedgypsum specimens and the tensile strength of the material:
�d = 10F
�D2
(2.84
t
D− 0.126
t
W+ 3.15
W
D+ 0.01
)−1(2)
where W is the height of the cylindrical section of the tablet definedin Fig. 1c. R represents the radius of curvature.
Eq. (2) gives the tensile strength of the material as function ofnormalised terms (t/D, W/D and t/W) which describe the shape.
The numerical values of the coefficients were determined fromexperiments carried out on discs with different curvature andthickness. In industrial R&D and manufacturing the break force oftablets is measured routinely thus Eq. (2) has significant practical
ig. 1. Diagram of diametrical compression test of thin flat discs: (a) round tablet,b) side view of round tablet with flat face, and (c) side view of round tablet withurved face.
pplications in relating the break force with a material propertynd is included in Pharmacopoeias (United States Pharmacopoeia,011).
For more complex tablet shapes, e.g. oval or elongated, there areo relationships available, although the equation developed by Pittased on experiments was extended to elongated shapes using aimple numerical procedure (Pitt and Heasley, in press).
An empirical relationship between the break force and tensiletrength of curved faced tablet manufactured from microcrystallineellulose powder was proposed by Shang et al. (2013):
t = F
�D2
(a
t
D+ b
W
D
)−1(3)
here the empirical parameters are a = 0.14 and b = 0.36. The formf Eq. (3) is based on Eq. (2), however, this simplified equationeduces to Hertz solution for flat face tablets and contains onlyhe minimum number of independent terms. The empirical equa-ions (2) and (3) can be used to relate the break force of curvedaced tablets to the tensile strength of the material; however theimitation of these equations is not fully established.
In the present paper we propose a numerical method to predicthe break force of tablets as an alternative to empirical approaches.umerical methods to model brittle failure exist (e.g. Li et al., 2011);owever, these models are complex and thus are unlikely to bedopted in industrial practice in their current form. Simplifications therefore needed. In this paper we make use of standard finitelement analysis and explore if simple failure criteria could be usedo predict the break force of tablets without resorting to fracture
echanics or advanced computational methods. A failure criterions selected and validated against experimental data. The conditionsor practical applicability are identified.
. Numerical modelling procedures
.1. Material
Numerical simulation is performed using a constitutive modelalibrated for microcrystalline cellulose Avicel PH102 (manufac-ured by FMC BioPolymer, Cork, Ireland). This powder material wassed to generate empirical data (Shang et al., 2013) which is usedor validating the numerical results presented below.
Microcrystalline cellulose is an excipient which is used widely
n pharmaceutical tablet formulations. The particles are irregular,
ith a nominal particle size of 100 �m and size distributionetween 20 and 200 �m according to the manufacturer’s specifi-ations. The loose bulk density of the powder is between 280 and
harmaceutics 445 (2013) 99– 107
330 kg/m3. Density is important in constitutive model calibrationwhich is described in detail in Section 2.3. For the loose bulk den-sity and the density of the fully densified material (grain density)the values of 300 kg/m3 is 1590 kg/m3 are used, respectively.
2.2. Geometry and loading
The finite element software ABAQUS/Standard (commercialisedby Dassault Systèmes) is used to simulate the diametrical compres-sion test. Three-dimensional (3D) static stress analysis is performedfor flat face and curved face tablets using 8-node linear brick type ofelements (C3D8). One quarter of the tablet is modelled for symme-try reasons in order to reduce the number of degrees of freedom.The platen is simulated as a rigid body and the interaction betweenthe platen and the tablet is considered frictionless. Compressionis simulated by applying displacement boundary conditions to theplaten.
The analysis of break force is done for the range of tablet shapespresented in Table 1. The choice of the geometric futures is dis-cussed in Section 3.2. The initial relative density (RD) distributionis assumed uniform in the tablet. RD is defined by the mass ofthe tablet divided by the total volume of the tablet (including thepores). The RD of the fully densified material is unity.
2.3. Constitutive model
Two constitutive models are used, in the first model the con-stitutive parameters are constant and in the second model theconstitutive parameters change according to RD:
1. Linear elastic model. The elastic properties (Young’s modulus Eand Poisson’s ratio �) determined experimentally are functionsof relative density (Shang, 2012) as shown in Fig. 2. The lin-ear elastic model has fixed parameters which corresponding tothe chosen value of initial RD. The data analysis procedures aresummarised elsewhere (Shang et al., 2012).
2. Elasto-plastic model. The elastic properties are considered as inthe elastic model above. The yield surface is described usingthe Drucker–Prager cap (DPC) model with density dependentparameters. The DPC model is widely used to simulate powdercompaction (PM Modnet Research Group, 2002; Brewin et al.,2008). The yield surface has two parts, a shear failure line, charac-terised by cohesion and internal friction angle and an elliptic capsurface. We use the DPC model as implemented in Abaqus, wherethe flow rule is associated on the cap surface and non-associatedfor the shear failure line. The model calibration procedures aredetailed elsewhere (Shang et al., 2012). Fig. 3 presents a family ofDrucker–Prager cap surfaces corresponding to different levels ofrelative density. The material parameters of the DPC model varywith the relative density. To describe elastic behaviour Young’smodulus and Poisson’s ratio are considered also dependent ofrelative density as presented in Fig. 2.
The break force of flat faced tablets is simulated for a range ofrelative densities RD = 0.6–0.94 (Table 1). For curved face tablets afixed value for the initial relative density of RD = 0.84 is selected, asthe geometry factor is the concern.
When the linear elastic model is used the parameters are fixed toE = 5.2 GPa and ratio is � = 0.3; these values correspond to a relativedensity of 0.84. For MCC this RD value is obtained for compressionpressures between 100 and 150 MPa which is within the practical
range. When the DPC model is used the initial density distribu-tion in the tablet is assumed uniform, thus the elastic and plasticmodel parameters are the same at every point in the body in the ini-tial configuration. This model however allows densification to take
C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107 101
Table 1Tablet geometry (see Fig. 1) and initial density for simulations.
Tablet geometry Tablet diameter (D) (mm) Radius of face (R) (mm) Band width (W) (mm) Thicknessa (t) (mm) Initial RD
Fig. 2. Elastic properties as functions of relative densit
lace due to the loading conditions and the material parametersre changing as function of relative density as illustrated in Fig. 3.hus if densification takes place then the yield surface expands asllustrated in Fig. 3 and the elastic modulus also increases as shownn Fig. 2a.
.4. Failure criteria
Three failure criteria are considered in order to obtain an esti-ate of the break force:
0
20
40
60
80
100
120
-50 0 50 10 0 15 0
Effec
�ve
stre
ss ,
MPa
Hydrosta�c stress , MPa
0.9
0.85
0.8
0.7
ig. 3. Drucker–Prager cap surfaces, the labels indicating relative density (Shang,012).
Rela�ve densit y
oung’s modulus and (b) Poisson’s ratio (Shang, 2012).
1. Criterion 1: Maximum principal stress, used with the linearelastic model. When the stress along the diameter reaches thematerial tensile strength �d the reaction force of the platen isregarded as the break force of the tablet.
2. Criterion 2: Maximum principal stress, used with the variableDPC model. This is same as criterion 1, however, it is treated asa separate case because it is used with a different constitutivemodel.
3. Criterion 3: Fully developed shear failure zone (used with thevariable DPC model). As the load is increased, elements betweenthe loading platens experience yielding on the shear failure line.The tablet is considered failed when the shear failure zone prop-agates through the centre of the tablet.
The tensile strength of the material depends on the relative den-sity as illustrated in Fig. 4, where the experimental data are fromShang et al. (2013). For example, for the density range employed inmodelling the break force of flat faced tablets the tensile strengthis in the range of 1.35–10.47 MPa; for RD = 0.84 and �d = 6.4 MPa.
3. Results and discussion
The analysis of the break force of flat and curved face tablets ispresented in separate sections below.
3.1. Flat faced tablets
Fig. 5 illustrates the application of the three failure criteria forthe analysis of the flat faced tablets with an initial relative den-sity of 0.84. Fig. 5a and b shows that the maximum principal stressstate within the tablet using linear elastic model and DPC model
102 C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107
0
2
4
6
8
10
12
14
0.2 0.4 0.6 0.8 1
Tens
ile s
tren
gth,
MPa
Rela�ve density
Criterion 1
Criterion 2
Criterion 3
Experiment
Ft
(mttpufpdoaitifom(tvr
iywT
0
50
100
150
200
250
300
350
0 0.140.12 0.10.080.060.040.02
Forc
e, N
Displacement, mm
Criterion 1
Criterion 2 & 3
DPC model
Linear elas�cmodel
ig. 4. Tensile strength determined using Eq. (1) vs. relative density for flat facedablets.
grey region), respectively. Fig. 5c highlights the area where ele-ents yield on the shear failure line (red region) developed along
he centre of the specimen. The local deformation around the con-act points leads to high stress areas which are not considered; inractice local crushing of the tablet is often observed; however,ltimately the tablet fails due to tensile stresses as discussed asollows. In Fig. 5(a), the first element which reaches the maximumrincipal stress appears at the centre of the tablet. In Fig. 5(b), theistribution of maximum principal stress suggests that the locationf the first element which reaches the material tensile strength ist somewhere in between the loading point and centre of the spec-men. The element where the maximum principal stress exceedshe material tensile strength is considered as the location of thenitiation of the crack. Modelling of diametrical compression of flataced tablets by Procopio et al. (2003) also showed that the locationf maximum principal stress was dependent on the constitutiveodel used and the prediction of an elasto-perfectly plastic model
which accounted to local flattening around the contact betweenablet and loading platen) was consistent with experimental obser-ations. (For interpretation of the references to color in this text, theeader is referred to the web version of the article.)
Using the above criteria the break force for flat face tablets
s established using the following method. The numerical anal-sis generates the force at the loading platen which is plottedith respect to the displacement of the platen as shown in Fig. 6.
he maximum principal stress distribution is examined. For the
Fig. 5. Diametrical compression test simulation of flat face tablet using
Fig. 6. Force displacement curve of 3 failure criteria at RD = 0.84 of flat faced tablet.
linear elastic constitutive model the break force is taken whencriterion 1 is satisfied. When the DPC model is used a differentforce–displacement curve is generated (Fig. 6) as the DPC modelallows plastic deformation at the contact point, which leads to localdensification; this is discussed below in more detail. The simula-tion generates the force–displacement curve of the loading platen.When failure criteria 2 or 3 are satisfied the corresponding displace-ment and force correspond to the breaking point as illustrated inFig. 6 using solid markers.
The predicted break force using the 3 failure criteria give approx-imately the same force value. When DPC model is applied the breakforce prediction using criteria 2 and 3 gives exactly the same resultfor this particular relative density. The predicted break force usingthe three failure criteria is similar for all densities. Knowing thebreak force the tensile strength is determined using the Hertz solu-tion (Eq. (1)).
As shown in Fig. 4, all three failure criteria are in good agree-ment with each other and with the experimental data. Tensilestresses develop not only in the centre of the tablet but also inthe region of contact between tablet and platen as shown in Fig. 5aand b. The flattening of the contact area is captured by the DPCmodel. In this area plastic deformation leads to further densification(Fig. 7a). Elements in this region reach the material strength before
the elements along the diameter where the initial crack initiates.Therefore the stresses near the contact region are not consideredwhen the failure criteria 1 and 2 are applied.
C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107 103
and (
opf
acoetdmp
fegtat
3
et(l
This normalisation procedure is consistent with Shang et al. (2013)to analyse experimental data for tablets of different geometriespressed to different densities.
Fig. 7. (a) Density at contact point
Along the centre of the tablet where tensile stresses are devel-ped and the material yields along the failure surface the modelredicts dilation because the flow rule is non-associated for theailure surface, these effects are also shown in Fig. 7a.
Fig. 7b shows the contact region for a tablet made of MCC, rel-tive density RD = 0.854 and thickness 2.285 mm. In addition toontact flattening one can observe secondary cracks, which are alsobserved by others experimentally (Rudnick et al., 1963; Jonsént al., 2007; Mates et al., 2008). Procopio et al. (2003), identifiedhat the location of crack initiation at a point along the loadingirection just under the densified zone. This is consistent with theodel prediction (Fig. 5b). The cracks originating in this zone will
ropagate to form a continuous failure plane across the diameter.Although the three failure criteria predict the same break
orce, the stress states between linear elastic and DPC mod-ls are different. Stress states become more complex as theeometry of the tablet is changed from flat to curved faces. Inhe next section we examine the applicability of the modelsnd failure criteria to predict the break force of curved facedablets.
.2. Curved faced tablets
The geometry of curved faced tablets is defined in Fig. 1. Thexperimental data used in the development of the empirical equa-
ion (3) is presented in Fig. 8 in the t/D–W/D space. Flat faced tabletsW = t) lie on the bisector of the plane. Convex faced tablets areocated at the right hand side of this line. The validity region of Pitt’s
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
W/D
t/D
Experiment data
Simula�on data
Limited region
Pi�'s region
Fig. 8. Simulation and experimental data.
b) photograph of failed specimen.
empirical relationship (Eq. (2)) is indicated in Fig. 8. The geometriesconsidered for numerical analysis (Table 1) are presented alongsidethe experimental data points in Fig. 8.
A limited region is highlighted in Fig. 8. The considerations usedto select this limited region are based on tablet shapes and failuremodes that are presented in Section 3.3.
In order to compare the break force of tablets of various shapesnormalisation is necessary. The normalised force is defined as:
F̄ = F
Fref(4)
where the reference force is
Fref = �d × �A
2(5)
where A is the cross-sectional area of the tablet which can be cal-culated from the geometric features in Fig. 1a:
A = D2
[2(
t
D− W
D
)(((t/D) − (W/D))2
16+ 1
3
)+ W
D
](6)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.00 0.05 0.10 0.15 0.20 0.25
Nor
mal
ised
forc
e
Displacement, mm
Criterion 1
Criterion 2
Criterion 3
DPC model
Linear elas�c mod el
Fig. 9. Normalised force displacement curve of 3 failure criteria at RD = 0.84 forcurved face tablet with a geometry (W = 1.575 mm, t = 2.91 mm).
104 C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107
F1
fesibtmwtf
fI
Table 2Coefficients for Eq. (3) for experimental data and the three failure criteria.
Relative density = 0.84 a b
Criterion 1 0.419 0.016
by Shang et al. (2013) are more complex than that of flat discsand can include delimitations as illustrated in Fig. 13. In Fig. 13aand b, besides the main crack across the diameter, cracks appearin the band width area near the loading platen. Such breakage
ig. 10. Surface fitted to simulation data points using 3 failure criteria: (a) criterion, (b) criterion 2 and (c) criterion 3.
The normalised force–displacement curve obtained numericallyor W = 1.575 mm and t = 2.91 mm is illustrated in Fig. 9 for the linearlastic model and the DPC model. The break force values corre-ponding to the three failure criteria are identified in Fig. 9. It isnteresting to observe that due to the normalisation method thereak force for the linear elastic model and criterion 1 can be higherhan unity. For the DPC model the force–displacement curve deter-
ined numerically is unique and the application of criteria 2 and 3ill give different values of the break force: criterion 2 corresponds
o first yield which occurs earlier than a fully developed plastic zone
or criterion 3.
The normalised forces using the 3 failure criteria are plotted asunctions of t/D and W/D in Fig. 10 for an initial density of RD = 0.84.n Fig. 10a, criterion 1 is applied and the normalised force have a
range of 0.9–1.2. In Fig. 10b, the predicted force using criterion 2 isrange from 0.6 to 1. In Fig. 10c an estimated force using criterion 3give almost constant force prediction around 1.
For each criterion a surface given by Eq. (3) is fitted as presentedin Fig. 10 (Table 2).
In order to compare the force prediction using 3 failure criteria,Pitt’s equation and the experimental data are plotted in Fig. 11.
Fig. 11 suggests that criterion 2 is closer to the experimental datathat the other criteria and Pitt’s equation. For a more detailed exam-ination the five surfaces are sectioned alongside the lines labelledin Fig. 8 and the curves are plotted in Fig. 12.
The differences between empirical models (Pitt, Shang) and thenumerical approach are relatively small for shallow curvatures andincrease progressively as the curvature is increased to standard,deep and extra deep shapes. It can be observed that criterion 2 isa better representation of the experimental data which is fit usingthe equation developed by Shang et al. (2013). It is also important tonote that the 3 failure criteria and the experimental data are fittedusing Eq. (3) which converges to the Hertz solution for flat facedtablets. The form of the Pitt’s equation (2) does not reduce to theHertz solution for flat faced tablet and its prediction is consistentwith Eq. (3) for relatively large t/D values and overestimates thebreak force for small t/D.
3.3. Tablet failure
A limited region of simulation data is presented in Fig. 8. Thislimited region is defined by 0.15 < W/D < 0.3 and t/D < 0.45 from thefollowing considerations:
1. For W/D > 0.3 the tablet is considered to have a band width whichis too wide for typical pharmaceutical tablets.
2. For t/D > 0.45 the aspect ratio is not consistent with typical phar-maceutical tablets (the thickness is too large). In addition forthese geometries the failure patterns observed experimentally
Fig. 11. Normalised force surfaces in 3D as functions of the geometric factors t/Dand W/D.
C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107 105
0
0.2
0.4
0.6
0.8
1
1.2
0.1 0.2 0.3 0.4 0.5 0.6
Nor
mal
ised
for
ce
t/D
Sha llow
ShangPi�Criteria 1Criteria 2Criteria 3
0
0.2
0.4
0.6
0.8
1
1.2
0.2 0.3 0.4 0.5 0.6
Nor
mal
ised
for
ce
t/D
Standard
ShangPi�Criteria 1Criteria 2Criteria 3
(a) (b)
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7
Nor
mal
ised
for
ce
t/D
Dee p
ShangPi�Criteria 1Criteria 2Criteria 3
0
0.2
0.4
0.6
0.8
1
1.2
0.4 0.5 0.6 0.7
Nor
mal
ised
for
ce
t/D
Extra Dee p
ShangPi�Criteria 1Criteria 2Criteria 3
(c) (d)
0
0.2
0.4
0.6
0.8
1
1.2
0.2 0.3 0.4 0.5 0.6
Nor
mal
ised
for
ce
t/D
Line labell ed (e) in Figure 4
Shang
Pi�
Criteria 1
Criteria 2
Criteria 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.2 0.3 0.4 0.5 0.6
Nor
mal
ised
for
ce
t/D
Line labell ed (f) in Figure 4
Shang
Pi�
Criteria 1
Cri teria 2
Criteria 3
(e) (f)
F Shangc p, (d)
3
ig. 12. Comparison between empirical equations and 3 failure criteria. The labels “urvatures used in the experiments are labelled as (a) shallow, (b) standard, (c) dee
(Addinall and Hackett, 1964) suggests that tensile stress distri-butions are different from the Hertz solution.
. For W/D < 0.15 the numerical model shows stress states that aremore likely to produce delaminations rather than breakage oftablet across the loading points as in the case of flat discs. Thestress analysis using the DPC model for the geometry D/R = 0.67,
W/D = 0.1 is provided in Fig. 15. The elements in the grey regionjust under the densification zone near the loading boundaryreach the material tensile strength in “Y” direction (break intolayers/laminate). The failure pattern of tablets after diametrical
” and “Pitt” refer to Shang et al. (2013) and Pitt et al. (1988), respectively. The punchextra deep, (e) line labelled (e) in Fig. 8 and (f) line labelled (f) in Fig. 8.
compression testing is examined non-destructively in 3D usingX-ray computed tomography (CT) using procedures describedelsewhere (Sinka et al., 2004). The failure patterns inside shal-low and ball shape tablets are presented in Figs. 15 and 16,respectively. It can be seen in the X–Y plane (Fig. 15b) andin an offset plane (Fig. 15d) that a crack is different from the
major crack in X–Z plane (Fig. 15a) and develops from theloading boundary. This is consistent with the simulation resultin Fig. 14 showing tensile stresses that lead to crack in thisdirection.
106 C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107
Fig. 14. Maximum principal stress analysis of D/R = 1.25, W/D = 0.1 at RD = 0.84 usingDPC model.
F(
eometry: t/D = 0.464, W/D = 0.198 and D = 10.347 mm; (b) compaction pressure:5 MPa, mass: 400 mg, geometry: t/D = 0.552, W/D = 0.151 and D = 10.387 mm.
Fig. 16 presents X-ray CT images of the failure patterns inside a
all shape tablet (W/D < 0.1 and t/D = 0.7–0.8, please see Fig. 8). Itan be seen that the tablet delaminates. The corresponding photosf this tablet are shown in Fig. 16a and b. The local flattening
behaviour of the tablet near the loading area and the fracture planecan also be observed. This failure mode suggests that the breakageof this shape involves more complex mechanisms than found in
thin flat discs.
and D = 10.322 mm) at different locations: (a) x–z plane; (b) x–y plane; (c) y–z plane;
C. Shang et al. / International Journal of Pharmaceutics 445 (2013) 99– 107 107
F ic iml
4
fTacm
sotflcsbbbo
iombdtu(tbw
nfc
A
G
215–224.
ig. 16. (a) Photos of breakage of ball shape tablet and X-ray computed tomographocations: (a) x–z plane; (b) x–y plane; (c) y–z plane.
. Conclusions
A numerical methodology to estimate the break force of curvedaced tablets under diametrical compression loading is proposed.he method does not involve fracture mechanics and can be easilypplied in industrial practice. The analysis can be performed usingommercial finite element software provided that the constitutiveodel of the material and the failure criterion are established.Linear elastic constitutive models with maximum principal
tress based criteria were found to over-predict the break forcef tablets due to complex material behaviour around the concen-rated loading points. Constitutive models that account for plasticattening around the loading points and possible densification inertain regions, such as the Drucker–Prager cap model with den-ity dependent material parameters are suitable to describe theehaviour of powder compacts. It was found that failure criteriaased on a fully developed plastic zone are also over-predicting thereak force while maximum principal stress based criteria at thenset of failure are in good agreement with the experimental data.
For materials that exhibit brittle behaviour Eq. (3) can be cal-brated using numerical experiments which can be used in theirwn right to relate the break force of curved faced tablets under dia-etrical compression to the tensile strength. This information can
e used directly in pharmaceutical formulation design and processevelopment. For complex tablet shapes it is necessary, however,o consider the failure patterns of the tablet. These observations aresed to define and restrict the applicability of the model obtainede.g. for tablets with deep curvatures and tablets with small bandhickness). The conditions whereby the maximum principal stressased criterion at the onset of failure gave consistent predictionsith the failure modes observed experimentally were identified.
Although validated for round tablets only, it is proposed that theumerical procedures developed may be used to predict the break
orce of oval and elongated shape tablets. This numerical analysisan replace empirical equations for more complex shapes.
cknowledgements
C. Shang acknowledges the Mechanics of Materials Researchroup at Leicester University for a partial PhD scholarship. The
ages of ball shape tablet (t/D = 0.721, W/D = 0.0538 and D = 10.344 mm) at different
paper was prepared while the corresponding author was on aca-demic study leave.
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