# Investigations into the tensile failure of doubly-convex cylindrical tablets under diametral loading using finite element methodology.Int J Pharm. 2013 Sep 15; 454(1):412-24.IJ

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 σt of doubly-convex tablets can be calculated from σt=(2P/πDW)(W/T)=2P/πDT 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 (σt=2P/πDW) 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/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 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.

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*International Journal of Pharmaceutics,*vol. 454, no. 1, 2013, pp. 412-24.

*Int J Pharm*. 2013;454(1):412-24.

*International Journal of Pharmaceutics*,

*454*(1), 412-24. https://doi.org/10.1016/j.ijpharm.2013.06.069

*Int J Pharm.*2013 Sep 15;454(1):412-24. PubMed PMID: 23834836.