Powder Technology 238 (2013) 169 175 Contents lists available at SciVerse ScienceDirect Powder Technology journal homepage: www.elsevier.com/locate/powtec Determination of the tensile strength of elongated tablets Kendal G. Pitt a,, Matthew G. Heasley b a Global Manufacturing and Supply, GlaxoSmithKline, Priory Street Ware SG12 0DJ, United Kingdom b Research & Development, GlaxoSmithKline, Park Road, Ware, SG12 0DP, United Kingdom article info abstract Available online 29 December 2011 Keywords: Tablet tensile strength Finite element analysis FEA Shaped tablet Oval tablet The tensile strength of a tablet is an important attribute as the tablet needs to be mechanically strong enough to withstand further handling such as film coating, packaging, transport and end-use by the patient, but to be weak enough to break apart in the human body and so release its contents. Mathematical solutions to calculate tensile strength are known for flat-faced and convex-faced circular tablets. The work described here aims to extend this knowledge to capsule-shaped and oval-shaped tablets by means of 2-dimensional (2D) and 3-dimensional (3D) Finite Element Analysis (FEA). The stress analysis showed that as the tablet was elongated from a standard circular tablet to become that of an extended tablet shape, the peak principal tensile stress reached a limiting value. This limiting value was reached as the ratio of the length to width dimensions exceeded 1.7:1, which encompasses most modern pharmaceutical tablets. In addition the stress analysis shows that this limiting value approximated to 2/3 that calculated for circular tablets. Hence for a convex-faced elongated tablet the calculation for tensile strength generated by Pitt et al. [1] would become: σ t ¼ 2 10P 3 πd 2ð 2:84D t 0:126 W t þ3:15w D þ0:01þ where σ t is the tensile strength, P is the fracture load, D is the length of the short axis, t is the overall thickness and W is the wall height of the tablet, as shown in Fig. 1c. The solution was then checked by applying it to commercial tablets of differing shapes to demonstrate its utility. 2011 Elsevier B.V. All rights reserved. 1. Introduction The strength of a compact can be defined simply in terms of the compressive force required to fracture a specimen across its diameter [2].In the pharmaceutical industry this is referred to as a hardness test.more complex shapes can also be crushed by this method. However the breaking load does not take into account either the dimensions and shape of the compact or the mode of failure. The conversion of a fracture load to tensile strength, which takes these factors into account, allows for ready comparisons to be made between samples of different shapes or sizes. The tensile strength of a tablet is an important attribute as the tablet needs to be mechanically strong enough to withstand further handling such as film coating, packaging, transport and end-use by the patient, but to be weak enough to break apart in the human body and so release its contents. Generally, a tensile strength greater than 1.7 MPa will usually suffice in ensuring that a tablet is mechanically strong enough to withstand commercial manufacture and subsequent distribution. Tensile strengths down to 1 MPa may suffice for small batches where the tablets are not subjected to large mechanical stresses [3]. The solution for finding a tensile strength for a plane-faced round specimen from the diametral compression test was independently Corresponding author. E-mail address: Kendal.5.pitt@gsk.com (K.G. Pitt). and simultaneously developed by Carneiro and Barcellos [4] in Brazil, and by Akazawa [5] in Japan. The test is referred to as the Brazilian or indirect tensile test as a tensile fracture is induced by compressive loading. The test is simple and easy to perform and has been widely used to determine the tensile strength of a variety of brittle materials such as concrete [6], coal briquettes [7], Gypsum [8] and lactose tablets [9]. A complete analytical solution exists for the stress state induced by the loads [10]. A more detailed analysis of the diametrical compression tests, where the effects of contact flattening and plastic material behaviour were considered, was presented by Procopio et al. [11]. These papers [6 11] give the expression for tensile stress (σ t ), determination in a flat-face compact as: σ t ¼ 2P πdt This theory was further developed for determination of the tensile strength of cylindrical convex-faced compacts by Pitt et al. [1]: 10P σ t ¼ πd 2 2:84 t D 0:126 t W þ 3:15 W D þ 0:01 ð2þ Both Eqs. (1) and (2) are listed in monograph 1217 of the United States Pharmacopoeia [12] and are in routine use in the pharmaceutical industry. Increasingly, however, shapes more complex than simple ð1þ 0032-5910/$ see front matter 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.12.060
170 K.G. Pitt, M.G. Heasley / Powder Technology 238 (2013) 169 175 Fig. 1. Tablet shape and dimensions. (a): For Eq. (1). (b): For Eq. (2). (c): For Eq. (4). cylinders are in use for oral tablets. For example oval and capsule shaped tablets are designed both to assist in product identification and to assist in ease of patient swallowing. To date though solutions have not been calculated for the determination of the tensile strength of these elongated tablets due to the complexity of the mathematical analysis required. In the work below FEA was used to determine the stress distribution when tablets are loaded diametrically. The FEA model was first validated by comparing with the accepted stress solutions for flat-faced and convex-faced circular tablets, prior to being applied to elongated tablets. The FEA model resulted in a simple conversion curve for comparing round tablets to elongated tablets. This conversion was then checked by applying it to commercial tablets of differing shapes to demonstrate its utility. 2. Finite element simulation 2.1. Finite element simulation: initial setup 2D and 3D FEA simulations were conducted using commercial software COMSOL Multiphysics 3.4 along with the COMSOL MEMS module (COMSOL AB, Stockholm, Sweden). Static plain strain analysis combined with a deforming mesh was performed on a 2D geometry in order to simulate the circular flatfaced tablet and compare to the known solution in Eq. (1). The geometry used can be found in Fig. 2a. Only a quarter of the geometry was modelled as the rest could be simulated by symmetry. The tablet is represented by a quarter circle (bottom of Fig. 2b) and loading was Fig. 2. (a): Whole geometry showing the complete system and symmetry lines with the section used in the simulation shown is shaded. (b): Geometry setup for the simulations.
K.G. Pitt, M.G. Heasley / Powder Technology 238 (2013) 169 175 171 Fig. 3. (a): Example mesh for the whole model, with a box showing the contact area. (b): Example starting mesh for the contact area before elastic deformation. (c): Example mesh for the contact area with the applied load, showing the elastic deformation and contact flattening of the tablet. performed by applying a pressure to the top face of the block (shown at the top of Fig. 2a). In the initial geometry, there is an infinitely small contact area between the tablet and the compression plate, which would result in a theoretical infinite pressure at this location. In order to remove the infinite pressure and make the simulation closer to reality, the model incorporated a deforming mesh, which enabled the software to simulate contact flattening due to elastic deformation. The material used for the compression plate was steel (Young's Modulus of 2 10 11 Pa). In the absence of reliable mechanical properties for a tablet, it was decided to initially model the tablet as an isotropic, elastic material, with a Young's Modulus of 2 10 6 Pa. The choice of Young's Modulus was based on an understanding that the bulk of the contact flattening would be deformation of the tablet rather than deformation of the compression plates, and therefore was chosen to be significantly smaller than that of steel. The choice of an isotropic, elastic material was based on photographs of fractured tablets from Fell and Newton [9], which do not show significant plastic deformation of tablets after undergoing this compression test. Both the tablet and the compression plate were modelled as having a thickness of 5 mm. The pressure applied to the top face of the compression plate was 1 10 4 Pa in the -y direction. This pressure is equivalent to a force of 0.5 N. The force was chosen to be small as it is sufficient to show the relative tensile stress for a given compressive force, without the model being affected by plastic deformation. The compression plate was also constrained to move only in the y direction. 2.2. Finite element simulation: meshing Example meshes can be found in Fig. 3. A triangular mesh was used, with higher resolution on all three edges of the quarter-tablet, and along the bottom-most edge of the steel compression plate than in the bulk of these materials. The maximum element size around these edges was 50 μm. The bulk of the tablet was set with a maximum element size of 200 μm and the bulk of the compression plate was set with a maximum element size of 400 μm. The resulting mesh had 9673 triangular elements. 3. Results and discussion 3.1. Finite element simulation: model verification using circular tablets The simulation was first validated by comparing the results with the accepted stress solutions for circular tablets. The value of most interest to us was the failure mode of the tablets in this test. The failure mode is tensile stress in the x (horizontal) direction. Fig. 4a shows tensile stress in the x direction, showing that the negative values are numerically much larger than the positive values. However, the negative values are showing compression, so the important values
172 K.G. Pitt, M.G. Heasley / Powder Technology 238 (2013) 169 175 Fig. 4. (a): Initial model, showing compression and tension in the x direction. (b): Initial model, showing positive stress in the x direction only. Hence this shows tension in the x direction, which is the failure mode in the diametral compression test. Future models show positive tensile stress only. (c): Initial model, but with a high Young's Modulus, low compression and high spatial resolution. (d): Initial model but with high spatial resolution. are those above zero, which can be seen in Fig. 4b. The peak tensile stress of the model described thus far was 6447 Pa (4sf). Inserting the same numbers into Eq. (1), we have P=0.5 N, D=10 mm and t=5 mm. Hence σ t =6366 Pa (4sf). These tensile stress values, while not identical are within 2% of each other. As it is possible that the small difference in outcome is due to the deformation of the tablet in the simulation, the simulation was repeated but with a higher Young's Modulus for the tablet, at 2 10 10 Pa. This value results in minimal deformation of the tablet and therefore shows the relative effect of the deforming mesh. As shown in Fig. 4c, the result looked extremely similar to the initial model, but the peak tensile stress was almost identical to the theoretical calculation of Eq. (1), at 6366 Pa (4sf). This shows that the deforming mesh was the cause of the discrepancy in results between the simulation and the theoretical calculation. However, for all later models, the lower Young's Modulus (2 10 6 Pa) was used, as some elastic deformation would be expected with the higher forces used in a real test of tensile strength. In order to further check the accuracy of the initial simulation, a third simulation was performed, at a much higher resolution. In this high resolution model, the maximum element size at the point of contact was set to 2.5 μm and all other parts of the mesh were refined significantly. The resulting mesh had 171,168 elements. The results of this high resolution model, which can be seen in Fig. 4d, were extremely similar to those of the initial model, but with a higher spatial resolution and less points that appeared anomalous. The peak tensile stress was identical, at 6447 Pa (4sf). This high resolution simulation showed that the original resolution had sufficient accuracy for all future models. Similar comparisons were carried out between the initial simulation and Eq. (2), setting t=w, ie a flat-faced tablet. With Eq. (2), its accuracy requires 0.1 W/D 0.3 [1]. Hence further simulations
K.G. Pitt, M.G. Heasley / Powder Technology 238 (2013) 169 175 173 Fig. 5. Extended capsule shapes showing positive tensile stress in the x direction, with D=10 mm. (a): L=12.5 mm. (b): L=18 mm. (c): L=25 mm. were generated for comparison with thicknesses of 1 mm, 2 mm and 3 mm. These new thinner models, and the initial simulation, which had a thickness of 5 mm (W/D = 0.5), compared favourably with Eq. (2), all giving results within about 15% of the calculated result, which is consistent with the margin of error discussed in [1]. This shows that, within a small margin of error, Eq. (2) can also be applied to flat-faced tablets. These simulations are not shown here as the key difference between them and the original simulation is the numbers. 3.2. Finite element simulation: elongated tablets Elongation of the tablet was carried out by adding a rectangle onto the bottom of the quarter circle of the tablet and recalculating. Rectangles were added in 0.25 mm increments, which were equivalent to adding 0.5 mm of length to the whole tablet, while keeping the Peak tensile stress, Pa 6500 6000 5500 5000 4500 4000 1 1.5 2 2.5 3 Length / Width ratio Fig. 6. Effect of shape on tensile strength for a tablet. diameter of the semicircle that makes up the end of the tablet the same. The physics, mesh constraints and underlying equations were kept identical to the initial model, with the compression loading being along the longer axis. Fig. 5 shows the impact of lengthening the body of the tablet on the tensile stress distribution. The peak tensile stress from each simulation was added to a graph for comparison, as shown in Fig. 6. As the ratio of length to diameter was increased the stress reached a limiting value 2/3 that of a circular tablet, as shown in Fig. 6. This limiting value occurring at a length to width ratio of greater than 1.7:1; this encompasses most modern tablets. Hence for a flat-faced elongated tablet the calculation for tensile strength would become: σ t ¼ 2 2P ð3þ 3 πdt Similarly, for a convex-faced elongated tablet the calculation for tensile strength derived by Pitt et al. [1] would become:! σ t ¼ 2 10P 3 πd 2 2:84 t D 0:126 t W þ 3:15 W D þ 0:01 ð4þ 3.3. Finite element simulation: 3D The same simulation was extended to three dimensions. 3D simulations of tablets with flat faces agreed with the 2D simulations of the same and with Eq. (1) (for a circular tablet) or Eq. (3) (for an elongated tablet). 3D simulations of elongated tablets with convex sides (as shown in Fig. 1c) and of an unconventional tablet design having a different width at the centre than the diameter of the semicircle at the end of the tablet all agreed with Eq. (4), within a small margin of error as discussed in [1]. The 3D simulations had smaller compression plates and lower spatial resolution than the 2D models in order to minimise computation
174 K.G. Pitt, M.G. Heasley / Powder Technology 238 (2013) 169 175 Fig. 7. 3D model of a circular tablet under compression, showing positive and negative tensile stress in the x direction. requirements. However they did agree with the 2D models and the equations. An example 3D model is shown in Fig. 7. 3.4. Experimental data The validity of Eq. (4) was demonstrated by manufacturing two sets of tablets of identical formulation. One set was flat-faced circular tablets of 6 mm diameter and 100 mg compression weight. The other set was convex-faced oval tablets of 17 mm long by 7 mm wide and 750 mg compression weight. Both sets of tablets were compressed over a range of compaction pressures and then fractured to determine their tensile strength. Fig. 8 clearly shows agreement between the tensile strength calculated from Eq. (1) for the circular tablets and the tensile strength calculated from Eq. (4) for the oval tablets. It should be additionally noted that this was despite a 7-fold difference in volume between the tablet shapes. 3.5. Application to manufacturing During the development of a commercial tablet, product was compressed on a rotary tablet press both in a convex-faced round tablet shape and in a capsule-shaped tablet. On further processing, edge erosion was seen with the round tablets but not with the capsule shape. The round tablet hardness (compressive fracture load) was 60 to 90 N and the capsule shape hardness was 100 to 140 N. Converting these hardness values to tensile strength using Eqs. (2) and (4) respectively enabled the cause of the edge erosion to be assessed. The hardness values were equivalent to a tensile strength of below 1 MPa for the damaged round tablets and above 1 MPa for the capsule shape (Fig. 9), indicating that low tensile strength of the round tablets was the cause of the damage. 4. Conclusion The tensile strength of elongated tablets that do not show significant plastic deformation can now be readily estimated, so facilitating Fig. 8. Tensile strength comparison of flat-faced and oval convex-faced tablets. Fig. 9. Tensile strength comparison of tablet shapes.
K.G. Pitt, M.G. Heasley / Powder Technology 238 (2013) 169 175 175 the development and transfer of formulations and processes between differently shaped tablets. List of symbols σ t P D L t W References tensile strength [MPa] fracture load [N] length of short axis (equivalent to disc diameter) [m] length of long axis [m] overall thickness [m] tablet wall height [m] [1] K.G. Pitt, P. Stanley, J.M. Newton, Tensile Fracture of doubly convex cylindrical discs under diametral loading, Journal of Material Science 28 (1988) 2723 2728. [2] J.E. Rees, E. Shotton, Some observations on the ageing of sodium chloride compacts, Journal of Pharmacy and Pharmacology 22 (1970) 17S 23S. [3] D. McCormick, Evolutions in direct compression, Pharmaceutical Technology 17 (4) (2005) 52 62. [4] F.F.L. Carneiro, A. Barcellos, Tensile strength of concrete, RILEM Bulletin 18 (1953) 99 107. [5] T. Akazawa, RILEM Bulletin 16 (1953) 11 23. [6] P.J.F. Wright, Comments on an indirect tensile test on concrete cylinders, Magazine of Concrete Research 7 (1955) 87 96. [7] R. Berenbaum, I. Brodie, Measurement of the tensile strength of brittle materials, British Journal of Applied Physics 10 (1959) 281 287. [8] E. Addinall, P. Hackett, The effect of platen conditions on the tensile strength of rock-like materials, Civil Engineering and Public Works Review 59 (1964) 1250 1253. [9] J.T. Fell, J.M. Newton, Determination of tablet strength by diametral compression test, Journal of Pharmaceutical Sciences 59 (1970) 688 691. [10] S. Timoshenko, J.N. Goodier, Theory of elasticity, 2nd Edn McGraw Hill, New York, 1970. [11] A. Procopio, A. Zavaliangos, J.C. Cunningham, Analysis of the diametrical compression test and its applicability to plastically deforming materials, Journal of Materials Science 38 (2003) 3629 3639. [12] The United States Pharmacopeia 2011. 34th Edn. US Pharmacopeia Convention, Rockville, Maryland.