Investigations into the tensile failure of doubly-convex cylindrical tablets under diametral loading using finite element methodology

International Journal of Pharmaceutics 454 (2013) 412–424

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Investigations into the tensile failure of doubly-convex cylindrical tablets under diametral loading using finite element methodology Fridrun Podczeck ∗ , Kevin R. Drake, J. Michael Newton Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK

a r t i c l e

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Article history: Received 24 May 2013 Received in revised form 24 June 2013 Accepted 27 June 2013 Available online 5 July 2013 Keywords: Doubly-convex cylindrical tablets Diametral compression test Finite element method Tablet tensile strength

a b s t r a c t In the literature various solutions exist for the calculation of the diametral compression tensile strength of doubly-convex tablets and each approach is based on experimental data obtained from single materials (gypsum, microcrystalline cellulose) only. The solutions are represented by complex equations and further differ for elastic and elasto-plastic behaviour of the compacts. The aim of this work was to develop a general equation that is applicable independently of deformation behaviour and which is based on simple tablet dimensions such as diameter and total tablet thickness only. With the help of 3D-FEM analysis the tensile failure stress of doubly-convex tables with central cylinder to total tablet thickness ratios W/D between 0.06 and 0.50 and face-curvature ratios D/R between 0.25 and 1.85 were evaluated. Both elastic and elasto-plastic deformation behaviour were considered. The results of 80 individual simulations were combined and showed that the tensile failure stress  t of doubly-convex tablets can be calculated from  t = (2P/DW)(W/T) = 2P/DT with P being the failure load, D the diameter, W the central cylinder thickness, and T the total thickness of the tablet. This equation converts into the standard Brazilian equation ( t = 2P/DW) when W equals T, i.e. is equally valid for flat cylindrical tablets. In practice, the use of this new equation removes the need for complex measurements of tablet dimensions, because it only requires values for diameter and total tablet thickness. It also allows setting of standards for the mechanical strength of doubly-convex tablets. The new equation holds both for elastic and elasto-plastic deformation behaviour of the tablets under load. It is valid for all combinations of W/D-ratios between 0.06 and 0.50 with D/R-ratios between 0.00 and 1.85 except for W/D = 0.50 in combination with D/Rratios of 1.85 and 1.43 and for W/D-ratios of 0.40 and 0.30 in combination with D/R = 1.85. FEM-analysis indicated a tendency to failure by capping or even more complex failure patterns in these exceptional cases. The FEM-results further indicated that in general W/D-ratios between 0.15 and 0.20 are favourable when the overall size and shape of the tablets is modified to give maximum tablet tensile strength. However, the maximum tensile stress of doubly-convex tablets will never exceed that of a flat-face cylindrical tablet of similar W/D-ratio. The lowest tensile stress depends on the W/D-ratio. For the thinnest central cylinder thickness, this minimum stress occurs at D/R = 0.50; for W/D-ratios between 0.10 and 0.20 the D/R-ratio for the minimum tensile stress increases to 0.67, and for all other central cylinder thicknesses the minimum tensile stress is found at D/R = 1.00. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Pharmaceutical tablets are manufactured in a great variety of shapes. The mechanical strength of tablets is an important issue governing, for example, further processing such as film-coating, packaging and encapsulation. Sufficient mechanical strength is also required to ensure that tablets are fit for their purpose, can be handled without detriment and reach the patients in good, undamaged condition. The doubly-convex cylindrical shape is

∗ Corresponding author. Tel.: +44 020 7697 7178. E-mail addresses: [email protected], f [email protected] (F. Podczeck). 0378-5173/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpharm.2013.06.069

preferred over a flat tablet shape when tablets are used as intermediates in film-coating processes, or when tablets are filled into hard capsules. It was suggested that 70% of all pharmaceutical compacts are of doubly-convex shape (Kadiri and Michrafy, 2013). A great variety of methods to test the mechanical strength of tablets exists and has been reviewed recently (Podczeck, 2012). The USP35/NF30 (2011) provides details on suitable tests and test conditions for the mechanical strength of tablets and also suggests that flat cylindrical and doubly-convex tablets should be subjected to a diametral compression (indirect tensile) test, and the breaking load should be converted into a value of tensile strength. For flat cylindrical and doubly-convex tablets the USP35/NF30 monograph (2011) provides the necessary equations.

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List of symbols D PE R W T Im

tablet diameter photoelasticity radius of curvature of the convex part of the tablets central cylinder thickness total tablet thickness factor describing the average reduction in tensile stress due to face-curvature tensile strength first dimension in 3D-FEM simulations second dimension in 3D-FEM simulations third dimension in 3D-FEM simulations fitting parameter (Eq. (1.3)) fitting parameter (Eq. (1.3))

t X Y Z a b

For the flat cylindrical tablets, the Brazilian test (Barcellos, 1953; Carneiro, 1953; Fell and Newton, 1968) is usually employed, and its merits and shortcomings have been discussed recently (Podczeck, 2012, 2013). The tablets are fractured in diametral compression, and the analytical solution for the stresses causing failure resulted in the following equation used to calculate the tensile stress  t : t =

2P DW

(1.1)

where P is the breaking load [N], D [m] the tablet diameter and W [m] the cylinder thickness. In order to be able to use this equation, tablets must fail in tension i.e. the test conditions must ensure that the tablets break diametrally into two equal halves (Podczeck, 2012); crushing and other deviating failure patterns are not permissible (Newton et al., 1971). As there will be load spreading directly underneath the loading platens due to minute flattening of the tablets (Podczeck, 2012), care has to be taken to adjust the experimental conditions so that the resulting contact area does not exceed 10% of the tablet diameter (Timoshenko and Goodier, 1987); values up to 25% have been found acceptable in some special cases (Peltier, 1954; Marion and Johnstone, 1977). Work to develop an equation for the tensile strength of doubly-convex cylindrical tablets started with photoelasticity experiments (Pitt et al., 1989a), but the set of results, although providing some important potential relationships between tablet dimensions and tensile stresses, was too limited to allow the derivation of a final equation. Pitt et al. (1988) continued their work and derived an empirical equation from a set of data obtained on gypsum discs by statistical means. They stated that this equation was valid for brittle materials, and tablet dimensions with a W/D-ratio between 0.06 and 0.30 and face-curvature ratios D/R between 0.00 and 1.43 (for W/D = 0.06, D/R < 1.00), whereby R is the radius of the face curvature (Fig. 1). (Although the paper had appeared in print in 1988, it had been written after the photoelasticity paper, which had appeared in print in 1989; the earlier date of publication had been the result of different processing speeds by different journals.) The equation: t =



−1

10P T T W 2.84 − 0.126 + 3.15 + 0.01 D W D D2

(1.2)

was later used in a number of publications (e.g., Pitt et al., 1989b, 1990; Newton et al., 2000) and is the equation stated in the USP35/NF30 monograph (2011). Recently, Shang et al. (2013a) criticized the above equation claiming severe shortcomings on the following grounds: (1) Of the three dimensionless terms only two are independent and hence the equation should be simplified; (2) there is no scope to validate the equation further; (3) the equation does not reduce to the standard Brazilian equation (Eq. (1.1)) for W = T.

Fig. 1. Front and side elevation of doubly-convex tablets.

These authors carried on to develop their own equation: t =



P T W a +b D D D2

−1 (1.3)

where a and b are fitting parameters (see below). They claimed that their equation is superior to Eq. (1.2). Unfortunately, their equation suffers similarly from shortcomings. (1) They claim that the new equation is simpler. While it has only two independent dimensionless terms, it now has the two fitting parameters a and b, for which they provided various solutions (Shang et al., 2013b), and all parameters have been derived from experiments using compacts made from microcrystalline cellulose. There is some debate about microcrystalline cellulose being a material for which Eq. (1.1) is invalid (Procopio et al., 2003) due to elasto-plastic deformation, and the choice of this material in work to derive a new equation is hence questionable, although there is some evidence that microcrystalline cellulose tablets can behave elastic or elastoplastic during the diametral compression test depending on the test conditions, and elastic behaviour has been documented (Mashadi and Newton, 1988; Sinka et al., 2004). (2) Although the authors claim that their new equation reduces to the standard Brazilian equation for flat tablets, if W = T, this is in practice incorrect. In their first paper (Shang et al., 2013a) the coefficients a and b in Eq. (1.3) have a total value of a + b = 0.484 instead of the required total value of 0.5, which means that the tensile strength of flat tablets using this equation is overestimated by 3.3%. This is a rather large error, and Shang et al. (2013a) chose not to provide any data for flat tablets, comparing results obtained from Eqs. (1.1) and (1.3). In

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their further work (Shang et al., 2013b) they refined the values for the parameters a and b using finite element methodology, and the best refinement solution (criterion 2) resulted in values of 0.187 and 0.284, respectively, i.e. a + b = 0.471. This increases the discrepancies between the tensile stress of flat tablets (Eq. (1.1)) and the values obtained from Eq. (1.3) further to 6.2%. Overestimation of tensile failure stresses can be problematic, leading to unexpected problems during processing, packaging and handling of compacts. (3) Eq. (1.3) still requires the determination of three tablet dimensions, i.e. diameter, total and central cylinder thickness. The determination of the central cylinder thickness is troublesome and requires more sophisticated methods such as image analysis, which are not routinely available in, for example, tablet production, where the determination of the tablet tensile strength would be crucial. The need to determine the central cylinder thickness of doublyconvex tablets is one reason, why the pharmaceutical industry does not employ these calculations routinely, but chooses to rely on the determination of breaking loads only. Therefore, from a practical point of view Eq. (1.3) is not more helpful than Eq. (1.2). As Eq. (1.3) has been developed partly on the basis of experimental data for microcrystalline cellulose compacts and has the potential to overestimate tensile failure stresses grossly, the aim of the current work was to develop a general equation that is applicable independently of material and deformation behaviour of the compacts, and which is based on simple, readily measurable tablet dimensions such as diameter and total tablet thickness only. The resulting equation should fully converge into the standard Brazilian equation for a flat cylindrical tablet, if W = T and D/R = 0.00 (i.e. flat tablet). With the help of 3D-FEM analysis the tensile failure stress of doubly-convex tables with central cylinder to total tablet thickness ratios W/D between 0.06 and 0.50 and face-curvature ratios D/R between 0.25 and 1.85 were evaluated. Both elastic and elastoplastic deformation behaviour of the specimens was assessed. 2. Materials and methods 2.1. Materials Anhydrous dextrose was kindly provided by Rank Hovis Ltd. (High Wycombe, UK), and xanthan gum was kindly donated by A & E Connock (Fordingbridge, UK). Magnesium stearate was purchased from BDH (Poole, UK) and spray-dried lactose from Borculo Whey (Zeparox® , Needsweg, Netherlands). 2.2. Software Standard finite element method (FEM) was used (Abaqus 6.12.3, Dassault Systèmes, Vélizy-Villacoublay, France). Statistical data analysis was performed using SPSS 20 (IBM, Woking, UK), and cubic-spline interpolations were made using a Microsoft® approved add-on to Excel 2007 (SRS1 Software, Boston, MA). 2.3. FEM model description A 3D FEM model was used to model flat and convex tablets under diametral loading. The front and side elevation of the convex tablets and their relevant dimensions are shown in Fig. 1. Since the problem of convex discs is tri-planar symmetrical, only one eighth of the tablets was used in the models, and a similar approach was used for the flat tablets. Fig. 2 shows the eighth of a convex tablet, loaded via a stainless steel block, with all relevant symmetric boundary conditions applied. The use of only an eighth of the tablets in the modelling process is not only advantageous in reducing the computational effort (required memory and disc space; running times); it also permits the use of symmetric boundary conditions to prevent tilting, slipping, sliding and twisting of the model shapes

Fig. 2. One eighth of a doubly-convex tablet as implemented in 3D-FEM simulations. The load is applied as pressure via a steel block and is indicated by arrows. Boundary conditions are applied to the portion of the tablet in all three dimensions of space to prevent tilting, sliding or rotation during loading, and the boundary conditions applied to the steel block permit downward movement only.

when loaded. 3D-quadratic block elements (“20-node brick elements”; C3D20R: 3D stress analysis with reduced integration) were employed using a mapped meshing technique, and the number of elements required per shape was optimized by checking manually for convergence; the number of elements was increased until the changes in central (coordinates x,y,z = 0,0,0) and mean tensile stress along the z-axis were less than 0.001 MPa (<0.1%). Typically, this resulted in 51 nodes along the x-axis, 49 nodes along the y-axis and increasing numbers of nodes (7–33) along the z-axis depending on the total tablet thickness. In order to achieve comparable node numbers along the z-axis, cubic-spline interpolation was used during post-processing of the FEM results. The load was transmitted through the stainless steel block (dimensions of 1/4 block: L × B × H = 0.021 × 0.021 × 0.005 m; linear elastic isotropic engineering steel; Young’s modulus 209 GPa; Poisson’s ratio 0.3) to simulate real-life test conditions. The contact between block and tablets was considered as frictionless in tangential, and hard in normal direction, and a penalty approach was used to control surface penetration. For ease of modelling, smallscale sliding was assumed and node-to-surface elements were used so that the nodes of the steel block surface did not have to line up exactly with those on the tablet surface. For the tablets, three different test models were implemented: (1) a linear elastic model with the properties of Araldite CT200, hardened with 30% w/w Hardener 901, for which Young’s modulus of elasticity (2.58 GPa) and Poisson’s ratio (0.35) were taken from the literature (Burger, 1969). This model was chosen to enable a direct comparison with the photoelasticity work on doubly-convex cylindrical discs reported by Pitt et al. (1989a). (2) A linear elastic model with Young’s modulus of 9 GPa and Poisson’s ratio of 0.3 similar to work reported by Procopio et al. (2003); (3) an elastoplastic model with Young’s modulus and Poisson’s ratio as in model (1), plus a yield strength of 25.8 MPa at a plastic strain of 0.01. The model constants were chosen to reflect maximum underestimation of the failure stress in the simple Brazilian model as observed by Procopio et al. (2003).

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According to Stanley (2001) the Brazilian equation for the calculation of the tensile strength of a flat circular tablet (Barcellos, 1953; Carneiro, 1953; Fell and Newton, 1968) is valid up to a tablet thickness to diameter ratio W/D = 0.5, where W is the cylinder thickness and D is the diameter of the tablet. Photoelasticity work on doublyconvex tablets (Pitt et al., 1989a) evaluated W/D-ratios of 0.06, 0.1, 0.2 and 0.3, whereby the tablet diameter D was 0.05 m. In this work, taking various references into account (Pitt et al., 1989a; Shang et al., 2013a; Stanley, 2001), D = 0.05 m, and the W/D-ratios studied were 0.06, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4 and 0.5. These were combined with the following face-curvature ratios D/R in a 28 -design: 0 (“flat”), 0.25 (“micro”), 0.50 (“shallow”), 0.67 (“normal”), 1.00 (“unity”), 1.25 (“deep”), 1.43 (“coating”) and 1.85 (“ball”). For test model (1) all 64 combinations were simulated, whereas for models (2) and (3) only 8 combinations (W/D = 0.2 with all face-curvature ratios). Hence, a total of 80 FEM simulations were performed. The load was calculated to give standardized load intensity across the tablet central cylinder thickness i.e. 100 N/mm. This was converted into the following pressures: W/D = 0.06: 170,068 Pa; W/D = 0.1: 283,447 Pa; W/D = 0.15: 425,170 Pa; W/D = 0.2: 566,894 Pa; W/D = 0.25: 708,617 Pa; W/D = 0.3: 850,340 Pa; W/D = 0.4: 1,133,787 Pa; and W/D = 0.5: 1,417,234 Pa. These pressure values were applied on top of the steel block as indicated by small arrows in Fig. 2. 2.4. Preparation of tablets Two powder mixtures were prepared: (1) anhydrous dextrose with 1% w/w magnesium stearate, (2) xantham gum plus 47.5% spray-dried lactose and 1% w/w magnesium stearate. The powders (250 g per batch) were mixed in a 500 ml glass jar in a Turbula mixer (Type T2C, Willy Bachofen, Basel, Switzerland) for 20 min, whereby magnesium stearate was blended in for 3 min only in a post-mixing step. All tablets were prepared by direct compression on an instrumented tablet press (F-Press, Manesty, Liverpool, UK). For the dextrose tablets, 12.7 mm diameter deep concave punches were used and the tablets were produced at a rate of 60 min−1 at various compaction pressures, whereas the xantham gum tablets were pressed individually using 12.5 mm diameter punches of various concavity to achieve pre-defined levels of overall porosity. After tablet ejection, all tablets were weighed to ±0.001 g (Sartorius BP 121S, Göttingen, Germany), and the diameter and overall thickness values were determined to ±0.001 mm using a digital micrometer (Mitutoyo, Kawasaki, Japan). The height of the central cylinder thickness was determined to ±0.001 mm using image analysis (Solitaire 512, Seescan, Cambridge, UK). The breaking load of the tablets in diametral compression was determined using a CT-5 strength tester (Engineering Systems, Nottingham, UK) at a test speed of 1 mm/min. All tablets failed into two equal halves i.e. in tension. The breaking load was recorded to ±0.005 kg. 3. Results and discussion 3.1. Linear elastic behaviour (FEM Model 1) 3.1.1. Comparison with photoelasticity data Pitt et al. (1989a) studied seven different discs of 0.05 m diameter and different face-curvature using photoelasticity and compared them with the results of one flat disc. They prepared their discs from Araldite CT200, which they cured adding 30% w/w Hardener 901. They loaded these discs at 130 ◦ C with a standardized force of 3.9 N/mm central cylinder thickness for 1 h and then “froze” the stresses that had developed. They performed stress

415

analysis in all three dimensions of space using a scattered light technique and equipment typical for the time period during which this work was undertaken. The tediousness of the methodology and time restraints probably restricted the number of discs tested. For a W/D-ratio of 0.1 they studied D/R-ratios of 0, 0.5, 1, 1.25 and 1.43. To assess the influence of W/D-ratio on the results, they kept D/R at 1.43 and only changed the W/D-ratio to 0.06, 0.2 and 0.3. They validated their approach using the data obtained from the flat disc by comparing the values of tensile stresses along the normalized x(2x/D) and y-axis (2y/D) with theoretical values for 2D-calculations taken from Frocht (1948). (2x/D and 2y/D describe the distance of the observation point from the centre of the disc i.e. at the centre these values are zero, while at the outer perimeter the values are 1.) They also normalized the tensile stress at any point along the axes by dividing with the maximum tensile stress that they had obtained in the centre of the disc (x,y,z-coordinates 0,0,0). The experimental and theoretical results matched very well. They then attempted to validate the extension of their approach into the third dimension by comparing their results for the flat disc with results obtained using a sphere (Frocht and Guernsey, 1952). Here, agreement between the results was only seen between 2y/D and 2z/T having values between 0 and 0.6. Above the value of 0.6 along the y- or z-axis, the experimental results underestimated the theoretical values. The values along the x-axis did not agree with the theoretical values. In order to derive at the splitting stress of the convex discs, Pitt et al. (1989a) normalized the stress values obtained, regardless of W/D-ratio, with the maximum tensile stress obtained from the flat disc with W/D = 0.1. Hence, at this point a similar approach was used for the FEM results, and the matching data are compared with those obtained by Pitt et al. (1989a) in Fig. 3. Only the data for the yand z-axis have been compared, because the splitting stress causing tensile failure will originate from here. As can be seen, there is poor agreement between the photoelasticity and the FEM results. When comparing the results for W/D = 0.1 (Fig. 3A and C), at least the tensile stress at the centre of the discs appears similar, except for D/R = 1.43. However, the large peak stresses at 2y/D = 0.4 that photoelasticity data imply along the y-axis, particularly for D/R = 0.5 and 1.0 and less so for D/R = 1.25 are not seen with FEM. However, the FEM stresses gradually increase along the y-axis, having a maximum at 2y/D ≈ 0.85, followed by a sharp drop due to compressive stresses developing directly underneath the loading block. Along the z-axis photoelasticity stresses reduce quickly, whereas using FEM these stresses remain almost constant. When comparing the results for D/R = 1.43 and varying W/D-ratio (Fig. 3B and D), the central tensile stresses only match for the thinnest disc. Photoelasticity data exceed those found with FEM when comparing the z-axis and to some degree also when comparing the y-axis. There are hence considerable discrepancies between the two methodologies. The reasons for the differences must be found in the shortcomings of both methods. Price and Murray (1973) studied the diametral compression test of flat polymer discs with both methods and found a good agreement between the data, when points around the centre of the discs were compared, as was found in this study. However, away from the centre of the discs Price and Murray (1973) found that the two methods resulted in significantly different results, again as found in this study. Photoelasticity accuracy depends on the accuracy of the specimen dimensions, the casting technique, the slicing technique and accuracy, the heating temperature and time for which this temperature was maintained (Khayyat and Stanley, 1978). Pitt et al. (1989a) machined all discs from one and the same casting and kept the dimensions of the discs with 1 mm of the model shape. However, they heated all shapes at 130 ◦ C for one hour, regardless of the total thickness of these specimens, which might have resulted in variations in the fringe values. Due to the tediousness of the work, only 4 (y-axis) or less (z-axis) stress values were obtained, while the FEM results are based on 49 (y-axis)

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Fig. 3. Comparison of tensile stress results from 3D-FEM analysis with photoelasticity data reported by Pitt et al. (1989a). (A) z-Axis, W/D = 0.10; (B) z-axis, D/R = 1.43; (C) y-axis, W/D = 0.10; (D) y-axis, D/R = 1.43.

and up to 33 (z-axis) values. A deviation of Poisson’s ratio from a value of 0.5 will also contribute to inaccuracies associated with 3D-photoelasticity approximations (Frocht and Guernsey, 1952). For the material used Poisson’s ratio is 0.35 (Burger, 1969). On the other hand, FEM results depend on the density, shape and flexibility of the elements used, the contact model, material properties, boundary conditions etc. (MacDonald, 2007). However, the theoretical tensile strength for the flat discs, obtained using Eq. (1.1), is 1.273 MPa, and with 1.272 MPa (W/D = 0.06) the FEM result is only 0.1% off target, plus a further increase in element number did not improve the solutions. Convergence analysis confirmed that this solution could already have been obtained with half the number of elements, and did not improve by adding further elements. The use of a 20-node brick element is advantageous when modelling curved surfaces, because highly accurate solutions can be obtained with less elements (Fish and Belytschko, 2007), and the solutions are usually in very good agreement with the theoretical values (MacDonald, 2007). From the restrictions discussed above it cannot be concluded with certainty that the discrepancies between the two technical approaches are due to either or the other method. 3D-FEM has been used successfully in modelling stresses in cylindrical (Huang et al., 2012; Ruiz et al., 2000; Yu et al., 2006, 2009) and convex (Heasley and Pitt, 2009; Pitt and Heasley, 2013; Shang et al., 2013b) discs as well as flat tablets of various shapes (Drake et al., 2007), and photoelasticity has been successfully employed to determine the splitting stress of irregular shaped 3D-structures (Hiramatsu and Oka, 1966). 3.1.2. FEM results for flat discs of varying W/D-ratio Despite the differences between photoelasticity and FEM results further FEM modelling was carried out to find a valid stress solution enabling the calculation of the tensile strength of doubly-convex

tablets. Initially, a set of 8 flat discs with increasing W/D-ratio was processed. When using Eq. (1.1) for the calculation of the tensile strength of flat tablets, it is usually assumed that failure initiates in the centre (x,y,z-coordinate 0,0,0) of the specimen (Timoshenko and Goodier, 1987) and that stresses developing along the z-axis are zero or at least of no consequence unless W/D > 0.5 (Stanley, 2001). Hence, initially the tensile stresses developing at the centre of the discs were explored. Fig. 4A shows, that the central tensile stress declines in a polynomial fashion with increase in W/D-ratio. It is known that the Brazilian equation will render incorrect results for thicker discs (W/D > 0.5; Stanley, 2001) by not accounting for the larger tensile stresses developing along the z-axis (Yu et al., 2006). The polynomial function shown in Fig. 4A implies that in fact the results obtained using Eq. (1.1) always depend on the W/D-ratio, not only for tablets with a W/D-ratio above 0.5. The equation generally overestimates the tensile stresses developing in the centre of the discs unless the thickness of the discs is infinitely thin (convergence from a 3D- into a quasi-2D situation). For the set of flat tablets with the mechanical properties as described in Section 2.3, the following equation describes the reduction in central tensile stress with increase in W/D-ratio as seen in Fig. 4:

 W 2

t = −0.392

D

− 0.029

W  D

+ 1.273

(3.1)

(r2 = 1.000; RMS = 0.001%). However, this equation cannot be used to quantify the degree of deviation of the tensile failure stresses of flat discs from that obtained using Eq. (1.1) due to a clear influence of stresses developing along the z-axis on the tensile failure stress values, which is explored below. Fig. 5 shows the normalized tensile stress of the flat tablets as a progression of the position along the y- and z-axes. In order to compare the results, the stress values were normalized with the

F. Podczeck et al. / International Journal of Pharmaceutics 454 (2013) 412–424

Fig. 4. Tensile stress at the centre (A) and the central outer surface (B) of flat cylindrical tablets of different W/D-ratio.

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tensile stress obtained in the centre of each individual disc (x,y,zcoordinates 0,0,0). As can be seen, for the thinnest disc, along the y-axis (Fig. 5A) the tensile stress remains constant over approximately 65% of the central part of the axis. With increasing thickness, the stresses become less constant across the central part of the yaxis and the magnitude of this effect changes in the approximate order of 60%, 50%, 40%, 30%, and for all discs of W/D = 0.30 or more the stresses are constant only for about 20% along the y-axis from the centre of the disc towards the perimeter. In 2D-simulations, Fahad (1996) showed that due to load distribution over a finite area directly underneath the loading platens, the stresses along the y-axis were only constant up to 50% from the centre towards the periphery of the axis. Similarly, Hiramatsu and Oka (1966) reported that the tensile stress would be distributed across the y-axis uniformly only over the central part, just about half the diameter in width. In this work the above results show that when simulations account for all three dimensions of space, for the y-axis the portion across the central disc, which is under constant stress, depends on the W/D-ratio and can become much smaller than 50%. When the results for the z-axis are inspected (Fig. 5B), it can be seen that the tensile stress increases towards the outer surface of the discs, and this effect increases with increasing W/D-ratio. This is in agreement with Yu et al. (2006), who also found that with an increase in disc thickness the tensile stresses at the central outer surface (x,y,z-coordinates 0,0,1) of the discs increased. As seen from Fig. 5B, up to a W/D-ratio of 0.30 this increase in central outer surface tensile stress is comparatively small i.e. less than 10%, but for the remaining two thicknesses the increase is considerable. Fig. 4B shows the progression of central outer surface tensile stresses as a function of W/D-ratio. The polynomial function mirrors that shown in Fig. 4A with the mirroring axis being a line parallel to the x-axis at a tensile stress of 1.273 MPa. The polynomial function can be numerically described as follows:

 W 2

t = 0.812

D

+ 0.005

W  D

+ 1.272

(3.2)

As stated by Yu et al. (2006), the tensile failure stress of flat tablets will be equal to the tensile stress observed at the centre of the outer cylinder surfaces, not that seen in the central part of the flat tablets. This means that Eq. (1.1) systematically underestimates the tensile strength of flat tablets unless these are infinitely thin. From Eq. (3.2) it can be estimated that for a W/D-ratio of 0.15 the percentage underestimation is 0.5%. For W/D = 0.25 this value increases to 2.5%, for W/D = 0.3 the value is 4% and for W/D = 0.35 it is 5.8%. For even thicker discs the percentage underestimation exceeds 10%. In most practical situations the mechanical strength of tablets is optimized or maximized and hence underestimation is less critical, but there are examples, e.g. chewable tablets, where this should be borne in mind.

Fig. 5. Normalized tensile stress of flat cylindrical tablets of varying W/D-ratio as a function of the position along the y- (A) and z-axis (B).

3.1.3. FEM results for doubly-convex tablets In order to compare the stresses observed in the doubly-convex tablets with those of the flat tablets, the results need to be normalized. This could have been done as before i.e. as by Pitt et al. (1989a) by dividing all stresses at every point and for every disc with the theoretical tensile stress of 1.273 MPa. However, after inspection of Fig. 5 (see above) this did not seem justified, because not only is the tensile stress in the centre of each flat disc slightly different, but also the progression of the stresses along the z- and y-axes is different for each W/D-ratio. It was hence decided to normalize the stresses for each convex shape with the corresponding stresses of the flat tablets of similar W/D-ratio, using values matching exactly the same position along the z- (2z/T) or y-axis (2y/D). This normalization technique has been employed throughout the remainder of this investigation.

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Fig. 6. Normalized tensile stress of doubly-convex tablets of varying W/D-ratio as a function of the position along the z-axis. (A) D/R = 0.25; (B) D/R = 0.50; (C) D/R = 0.67; (D) D/R = 1.00; (E) D/R = 1.25; (F) D/R = 1.43; (G) D/R = 1.85.

Fig. 6 compares the normalized tensile stresses along the z-axis for the various face-curvatures. In all cases, the normalized tensile stresses are smallest for the thinnest central cylinder thickness (W/D = 0.06) and increase with increasing central cylinder thickness, whereby the rate of increase gets gradually less. Of great importance is, that except for W/D = 0.50 in combination with D/Rratios of 1.85 and 1.43 and for W/D-ratios of 0.40 and 0.30 in combination with D/R = 1.85 the normalized tensile stress increases slightly but consistently from the centre to the perimeter of the

tablets. This is similar to the progression seen when inspecting the profiles of the flat tablets (see above). For the named exceptions there seems to be a flip–flop effect i.e. here the normalized tensile stress decreases towards the outer surface of the tablets. As the opposite would have been expected this could be an indication of weaknesses in the tablets of excessive curvature combined with greater cylinder thickness and could, for example, be an indication of tendency for capping. This is further established in Fig. 7, which compares stress profiles for tablets with W/D = 0.20 with different

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“normal” doubly-convex tablets (D/R = 0.67) are the strongest and that hence this face-curvature ratio was recommended when maximization of the mechanical strength of a tablet formulation was sought. Pitt et al. (1990) found that the mechanical strength of acetylsalicylic acid tablets is at an optimum (maximum), when W/D = 0.2 and D/R = 0.25–0.67. It has to be borne in mind that the findings of these three research reports are based on Eq. (1.2). What all three research reports have in common with the current findings (Fig. 9A) is that a W/D-ratio of 0.2 appears to be favourable if mechanically strong doubly-convex tablets are to be developed. 3.2. FEM results for elastic and elasto-plastic behaviour

Fig. 7. 3D-FEM tensile stress patterns for tablets with different W/D- and D/R-ratios.

face-curvature (D/R = 0.00, 1.00 and 1.85) with a W/D = 0.50 tablet having a D/R-ratio of 1.85. The tablets with the W/D-ratio of 0.20 have stress patterns along the z-axis that indicate homogeneous stress distribution along the z- and y-axis (similar colour/grey profile), whereas the tablet with W/D = 0.50 shows that the stress profile varies considerably in both directions. Fig. 8 compares the normalized tensile stresses along the yaxis for the various face-curvatures. In all cases, the normalized tensile stresses are smallest for the thinnest central cylinder thickness (W/D = 0.06), but in contrast to the z-axis, the thickest tablets are not those with the highest tensile stresses. The tensile stresses increase only slightly from the centre towards the loading points, until a 2y/D-value of just over 0.8 is reached, at which the stresses suddenly increase rapidly to peak at a 2y/D-value between 0.87 and 0.88, and then to drop sharply. This progression of the tensile stresses along the y-axis is similar to that obtained for irregular-shaped rock samples (Hiramatsu and Oka, 1967). Above 2y/D-values of 0.9 compressive stresses dominate, as would be expected. The magnitude of the peak tensile stresses increases with an increase in face-curvature ratio D/R. Fig. 9A compares the normalized maximum tensile stresses for the different facecurvature ratios as a function of the W/D-ratio. For all D/R-ratios except D/R = 1.85, the largest peak value occurs at W/D = 0.20; for D/R = 1.85 the value at W/D = 0.15 is slightly larger, but in general it appears as though W/D-ratios of 0.15 and 0.20 are favourable when the overall size and shape of the tablets is modified to give maximum tablet tensile strength (maximum tensile stresses are regarded as proportional to the tensile strength of specimens, Yu et al., 2009). However, in no case is the maximum tensile stress of the doubly-convex tablets larger than that of a flat-face cylindrical tablet (Fig. 9B). The lowest tensile stress depends on the W/D-ratio. For the thinnest central cylinder thickness, this minimum stress occurs at D/R = 0.50; for W/D-ratios between 0.10 and 0.20 the D/Rratio for the minimum tensile stress increases to 0.67, and for all other central cylinder thicknesses the minimum tensile stress is found at D/R = 1.00. The findings seem to be in contrast to findings by Pitt et al. (1989b), who reported, using Eq. (1.2) for the calculation of the tensile strength of their tablets, that for a W/Dratio of 0.20 and larger the tablet strength would be practically independent of face-curvature. They also stated that for W/D = 0.1 the strongest tablets would be obtained with a D/R-ratio of 0.67. Newton et al. (2000) claimed, also using Eq. (1.2), that in general

Pitt et al. (1989a) found that for their limited set of photoelasticity data the normalized tensile stress in the centre of the tablets (x,y,z-coordinates 0,0,0) formed a linear relationship with the ratio of central cylinder thickness W to total tablet thickness T i.e. W/T. This relationship is explored in Fig. 10A for the full set of 80 FEM results. In general, the normalized tensile stress in the centre of the tablets follows a linear relationship with W/T, independent of whether an elastic or elasto-plastic model was used in the FEM simulations. As expected, there is also no difference between the two elastic models, which varied in Young’s modulus by a factor of 3.5 and between their Poisson’s ratios. Elasticity theory for linear elastic isotropic bodies predicts that stress distributions should be independent of Young’s modulus and Poisson’s ratio (Timoshenko and Goodier, 1987). However, discrepancies observed between photoelasticity and FEM results have triggered studies adding numerical proof to this theoretical principle, as far as FEM is concerned. That Poisson’s ratio does not influence the tensile stresses in 3D-FEM analysis of flat discs made from linear elastic materials was reported by Yu et al. (2009), and that larger differences in Young’s modulus equally do not affect the FEM results was demonstrated by Pitt and Heasley (2013). The current results demonstrate that these findings can be extended to doubly-convex shapes, as long as linear elastic materials are considered. In Fig. 10A some outliers are highlighted. The outliers are identical with the shapes that showed a flip–flop relationship for the normalized tensile stresses along the z-axis as discussed above (Fig. 6) i.e. except for W/D = 0.50 in combination with D/R-ratios of 1.85 and 1.43 and for W/D-ratios of 0.40 and 0.30 in combination with D/R = 1.85 the relationship between normalized tensile stress in the centre of the tablets and W/T-ratio is linear for W/D-ratios of 0.06 to 0.5. However, although it has been theoretically established by FEM analysis (Fahad, 1996; Ruiz et al., 2000) and practically shown on some tablets using high-speed video recording (Mashadi and Newton, 1988; Procopio et al., 2003) that at least for brittle flat cylindrical specimens failure is initiated at the centre of the discs, there is evidence that failure of shapes deviating from that of a flat cylinder is initiated at points between the tablet centre and the periphery (Hiramatsu and Oka, 1966). Pitt et al. (1989a) postulated that the point at which failure is initiated should be identical to the maximum tensile stress observed, which they termed “Imax ” and that the Imax -value was a factor with which the Brazilian equation (Eq. (1.1)) should be multiplied to derive at the tensile strength of the doubly-convex tablet. This did, however, not give realistic results (Pitt et al., 1988) despite the fact that the values of Imax were linearly related to W/T. Fig. 10B shows that the maximum normalized tensile stress is indeed linearly related to the W/T-ratio, but the progression of lines does not follow the W/D-ratio in a linear fashion. It would hence be difficult to find a common factor with which Eq. (1.1) could be multiplied, unless a more complex form similar to those derived by Pitt et al. (1988) and Shang et al. (2013a) was sought. In contrast to Pitt et al. (1989a), who claimed that the effect of the face-curvature on the tensile strength of doubly-convex

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Fig. 8. Normalized tensile stress of doubly-convex tablets of varying W/D-ratio as a function of the position along the y-axis. (A) D/R = 0.25; (B) D/R = 0.50; (C) D/R = 0.67; (D) D/R = 1.00; (E) D/R = 1.25; (F) D/R = 1.43; (G) D/R = 1.85. Compressive stresses (negative) were finite, but are omitted to avoid distortion of the graphs.

tablets could be incorporated into Eq. (1.1) by multiplying with Imax , Hiramatsu and Oka (1967) stated that this was only valid for ideal loading conditions. They pointed out that in practice ideal line loading of specimens did not occur, but that at the moment of failure the peak tensile stress was flattened due to local stress conditions and as a result of load spreading due to deformations

in the vicinity of the specimen-platen contacts. From their work they concluded that while the crack leading to failure did no longer originate in the centre of the specimens, an average stress along the failure plane should be used as factor to multiply the Brazilian equation (Eq. (1.1)), and that for the calculation of the average stresses along the y-axis the central 50%-portion only should be

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Fig. 9. Comparison of maximum tensile stresses occurring along the y-axis as determined by 3D-FEM. (A) Normalized maximum tensile stresses for different facecurvature ratios D/R as a function of the W/D-ratio of the tablets. (B) Maximum tensile stresses [MPa] for different W/D-ratios as a function of the face-curvature ratio D/R of the tablets.

used. The relationship between the average tensile stress along the central portion of the y-axis (50%) and the ratio W/T is shown in Fig. 11A. As before, some outliers had to be removed identical with the shapes that showed a flip–flop relationship for the normalized tensile stresses along the z-axis (see above). All other average stress values are linearly related to the W/T-ratio, and this function has a slope of 1.000 and an intercept of 0.000. The function is also maintained for the higher Young’s modulus model and the elastoplastic model i.e. is independent of the deformation properties of the tablets or the materials from which these tablets were formed. It seemed hence be interesting to check the validity of this relationship also for the stresses developing across the z-axis. In Fig. 11B this relationship is presented, whereby here 100% of the data were used to determine the average tensile stress. Again the relationship is fully linear, and there are no outliers, which lends further support to the assumption that it is the average tensile stress that causes failure, not the maximum stress. It can hence be concluded that for the determination of the failure stress of doubly-convex tablets the following simple equation provides theoretically correct approximations of the splitting stress leading to tensile failure: t =

2P Im DW

(3.3)

with P being the breaking load [N], D the diameter [m], W the central cylinder thickness [m], and Im = W/T, with T being the total thickness of the tablet [m] (Fig. 1). The equation can hence be simplified to t =

2P DT

(3.4)

421

Fig. 10. Normalized tensile stress along the y-axis as a function of the W/T-ratio for doubly-convex tablets with elastic or elasto-plastic behaviour. (A) Normalized tensile stress in the centre of the tablets; (B) normalized maximum tensile stress along the y-axis.

For a flat cylindrical tablet, T = W and hence Eq. (3.4) converts into the Brazilian equation. Eq. (3.4) is valid for doubly-convex tablets of central cylinder thickness to diameter ratio W/D between 0.06 and 0.50, except for W/D = 0.50 in combination with D/R-ratios of 1.85 and 1.43 and for W/D-ratios of 0.40 and 0.30 in combination with D/R = 1.85. Classical pharmaceutical tablets are unlikely to have such extreme dimensions. An exception are mini-tablets (i.e. tablets of 1–5 mm diameter), where the aim is to have equal dimensions in terms of total thickness and diameter. For mini-tablets neither Eq. (1.1) nor Eq. (3.4) are applicable. Eq. (3.4) applies to tablets that fail in tension independent as to whether they behave elastic or elasto-plastic under load. However, it has to be borne in mind that Eq. (1.1) underestimates the true tensile failure stress for elasto-plastic materials (Procopio et al., 2003). This will hence also apply to Eq. (3.4). In this work, the properties of the elastoplastic material had been chosen to result in a maximum reduction in tensile stress, compared to the elastic material. The tensile stress causing failure of the flat tablet (W/D = 0.20) with elastic properties was determined to be 1.252 MPa, whereas that with elasto-plastic properties was found to be 1.243 MPa, i.e. the discrepancy is about 1%. This was also observed for the various D/R-ratios, and the failure stresses are compared in Table 1. When tablet formulations are optimized, tablet tensile strength is usually not the only parameter of concern. While in most cases the strength will be a criterion to be maximized, it has to be compromised with, for example, tablet disintegration time or drug dissolution profile, which are given priority in optimization strategies. Underestimation of the tensile strength is hence not an issue; the opposite case of overestimation would be of much more concern. The only type of tablet where

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Fig. 11. Normalized average tensile stress as a function of the W/D-ratio for doublyconvex tablets with elastic or elasto-plastic behaviour. (A) Average tensile stress across the central 50% portion of the y-axis; (B) average tensile stress across the full z-axis.

underestimation of the tensile strength could be detrimental are chewable tablets, but these are not normally made from elastoplastic materials such as polymers or microcrystalline cellulose. Eq. (3.4) could also be used advantageously, if a doubly-convex tablet with defined tensile strength is sought. For example, if the target mechanical strength is that of a flat tablet of defined W/Dratio, but a tablet with normal face curvature (D/R = 0.67) and the same W/D-ratio as the flat tablet is to be developed, the breaking load of the flat tablet should be multiplied with the value of T/W to give the target breaking load of the doubly-convex tablet that would result in similar tensile strength values when compared with the flat tablet. Hence the new equation allows the setting of standards for the mechanical strength of doubly-convex tablets.

Table 1 Degree of underestimation of the tensile failure stress of elasto-plastic doublyconvex tablets with a W/D-ratio of 0.20 and various face-curvature ratios (D/R). D/R

Elastic

Elasto-plastic

% Deviation

0.00 0.25 0.50 0.67 1.00 1.25 1.43 1.85

1.252 0.953 0.763 0.668 0.529 0.449 0.400 0.312

1.243 0.946 0.757 0.663 0.525 0.446 0.398 0.311

0.73 0.77 0.81 0.82 0.68 0.61 0.53 0.32

Fig. 12. Relationships between tensile strength of tablets with different facecurvature ratios (D/R) and central cylinder to total thickness ratios (W/D), and various manufacturing parameters. (A) Influence of D/R-ratio of tablets with W/D = 0.20 and tablet porosity on the tensile strength; (B) tablet tensile strength as a function of compaction pressure for tablets with D/R = 1.25 and W/D-ratios between 0.14 and 0.21, prepared using 0.8 and 1.0 g of powder.

3.3. Experimental results The above discussion indicates that the normalized maximum tensile stresses of doubly-convex tablets for all D/R-ratios except D/R = 1.85 are largest at W/D = 0.20 and that in general it appears as though W/D-ratios of 0.15 and 0.20 are favourable when the overall size and shape of the tablets are modified to give maximum tablet tensile strength (Fig. 9A). It was also noted that doublyconvex tablets were always mechanically weaker than their flat counterparts (Fig. 9B). To obtain practical proof, 12.5 mm diameter tablets made of equal parts of xanthan gum and spray-dried lactose were prepared. The central core thickness was kept constant at 2.6 mm, resulting in a W/D-ratio of 0.2, and the total tablet thickness increased with increase in D/R-ratio. Three different tablet porosities were tested. As can be seen from Fig. 12A, similarly to the FEM simulations all doubly-convex tablets are weaker than the flat tablets and the lowest tensile strength was found for the D/Rratio of 0.5. An increase in D/R-ratio to a value of 1.0 increased the tensile strength significantly, but a further increase in the D/R-ratio to a value of 1.43 did not result in a statistically significant increase in tensile strength. The findings apply to all three values of tablet porosity. Fig. 9A implied that within a central core thickness to diameter ratio W/D between 0.15 and 0.20 for a given D/R-ratio the tablet tensile strength of doubly-convex tablets should be similar. This was experimentally tested preparing tablets from anhydrous dextrose with deep concave punches (D/R = 1.25). The increase

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in W/D-ratio was achieved by increasing the tablet weight from 0.8 to 1.0 g in combination with an increase in the compaction pressure and ranged from 0.14 to 0.21. The results are shown in Fig. 12B. If the implications derived from Fig. 9A are correct then the two different tablet weights should result in two similar functions of tablet tensile strength as a function of compaction pressure. As can be seen, the relationships between tensile strength and compaction pressure are linear (0.8 g: b = 0.0193; b0 = − 0.3784; r2 = 0.9874; 1.0 g: b = 0.0171; b0 = −0.3091; r2 = 0.9961). The 1.0 g tablets appear to have a slightly larger tensile strength, especially at larger compaction pressures. However, statistical testing to compare the two regression lines (Diem and Lentner, 1970) resulted in a non-significant difference (F = 1.82; F0.05 = 2.22; d1 = d2 = 18). Newton et al. (1971, 1972) pointed out that if two such regression lines are statistically the same, the tensile strength of the compacts can be regarded as similar. The results shown in Fig. 12B hence confirm that within the range of a W/D-ratio of 0.15–0.20 indeed the tensile strength of doubly-convex tablets is similar. 4. Conclusions With the help of 3D-FEM analysis the tensile failure stress of doubly-convex tables was evaluated and can be calculated from Eq. (3.4), i.e. the standard Brazilian equation (Eq. (1.1)) is multiplied with a factor Im = W/T. As a result, in Eq. (1.1) the cylinder thickness W is replaced by the overall tablet thickness T. Eq. (3.4) converts into the standard Brazilian equation (Eq. (1.1)) for flat cylindrical tablets when W = T, i.e. when the tablets are flat. The factor Im is valid both for elastic and elasto-plastic deformation behaviour of the tablets under load. The new equation is valid for all combinations of W/D-ratios between 0.06 and 0.5 with D/R-ratios between 0.00 and 1.85 except for W/D = 0.50 in combination with D/R-ratios of 1.85 and 1.43 and for W/D-ratios of 0.40 and 0.30 in combination with D/R = 1.85. FEM-analysis indicated a tendency to failure by capping or even more complex failure patterns in these exceptional cases. The most important practical implication of the new equation is that all required variable values can be obtained easily. It is no longer necessary to measure or calculate the central cylinder thickness of a doubly-convex tablet, removing the need for complex technologies such as image analysis or rough estimations. Instead, only the total tablet thickness is incorporated into the equation, which can be determined easily using digital or mechanical callipers. The new equation also allows the setting of standards for the mechanical strength of doubly-convex tablets by multiplying the target breaking load of a W/D-equivalent flat tablet with the ratio of 1/Im = T/W. The FEM-results further indicated that in general W/D-ratios between 0.15 and 0.20 are favourable when the overall size and shape of the tablets is modified to give maximum tablet tensile strength. However, the maximum tensile stress of the doublyconvex tablets will in no case be larger than that of a flat-face cylindrical tablet of similar W/D-ratio. The lowest tensile stress depends on the W/D-ratio. For the thinnest central cylinder thickness, this minimum stress occurs at D/R = 0.50; for W/D-ratios between 0.10 and 0.20 the D/R-ratio for the minimum tensile stress increases to 0.67, and for all other central cylinder thicknesses the minimum tensile stress is found at D/R = 1.00. Practical experiments did confirm these findings. Acknowledgements We are very grateful to David Bevan, Mark Iline and Dr Paul Fromme for the installation and maintenance of the Abaqus 6.12.3 software package. We also wish to thank Dr Kendal Pitt and

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