# Indices for the brittleness of pharmaceutical tablets: a reassessment

Brittleness is an important mechanical property. In the classical sense, a material is considered brittle if, during loading, it behaves elastically until failure. Nevertheless, it is also sometimes understood as the fact to be resistant to breakage. In the case of pharmaceutical tablets, three different indices have been defined to measure brittleness: the brittle fracture index (BFI), the brittle/ductile index (BDI) and the tablet brittleness index (TBI). The aim of this work was to reassess the meaning of the different indices that are known to give contradictory results. Using theoretical considerations, numerical modelling and experiments, it was possible to show that the only index that unequivocally measures the brittleness of the tablet understood as elastic until failure is the BFI. If the other two indices can be useful, for example to assess the friability of the tablet in the case of the TBI, they do not make it possible to measure tablet brittleness in the classical sense, i.e. as opposed to ductility.

## Introduction

Brittleness is an important property in the mechanics of materials like rocks, ceramics, metals or concrete, but also for pharmaceutical tablets. Despite the large use of the term, the definition of brittleness is not always very clear. Hucka and Das, among others, reviewed different definitions that can be found in the literature (Hucka and Das, 1974). Even if all the given definitions are not strictly the same, they all points toward the same direction which is that a solid is considered as brittle if, before failure, it does not present significant plastic deformation. As such, brittleness is considered as the opposite of ductility. In mechanical terms, brittle materials can be treated using the concepts of linear fracture mechanics, whereas ductile solids will be studied using the tools of plasticity and yielding. In the following text, we will refer to this definition of brittleness as “Elastic Until Failure”..

Generally, in the literature, brittleness is considered as a material property. Nevertheless, the problem is in fact more complex. Materials like rocks or pharmaceutical tablets are in fact quasi-brittle materials (Bažant, 2002, Girardot et al., 2023). This means that during a failure test, a fracture process zone (FPZ) develops ahead of a crack (micro-cracking, etc.) before the development of a macroscopic crack (Bažant, 2002). Depending on the material, the size of the FPZ can be very different. For example, FPZ in concretes are in general considered to have a size around some 100 mm (Hillerborg, 1985) whereas in a pharmaceutical tablet it can be only of a few mm (Girardot et al., 2023). The impact of the FPZ on the failure behavior in fact depends on the size of the structure. If the FPZ is very small compared to the typical size of the sample, the sample will behave like a brittle material whereas if the size of the FPZ is larger than the sample size, it will behave in a ductile way (Bažant, 2002, Morel and Dourado, 2011). Considering this definition, brittleness is the conjunction between a material property and the geometrical features of the test used.

One of the way to quantify this size effect, is to compare a characteristic length of the fracture process of the material to the actual size of the material. Hillerborg et al. proposed that the characteristic length of the fracture process of the material (l_{c}) can be expressed using the material properties as (Hillerborg et al., 1976):

With E the Young modulus of the material, G_{c} its fracture energy (i.e. the energy required to generate a unit area of a crack) and σ_{t} its tensile strength. The shorter l_{c} compared to the size of the structure, the more brittle the material will be. Equation 1 can be rewritten using the critical stress intensity factor K_{c}, which can be written as:

Equation 1 thus becomes:

In the present paper, we will only discuss cases where the failure occurs in mode I (tension), so we will consider G_{Ic} for the fracture energy and K_{Ic} for the critical stress intensity factor..

Beside this definition based on the opposition to ductility, Brittleness is sometimes defined based on the strength of the material. For example Rosato and al. stated that (Rosato and Rosato, 2003): “Brittleness identifies material easily broken, damaged, disrupted, cracked, and/or snapped”. This definition in fact relies more on the sense of brittle in the everyday life. From a fundamental point of view, it is completely different from the other definition. A perfectly brittle solid in the previous sense (i.e. a solid that can be treated with the tools of linear fracture mechanics) would resist more or less to a crack propagation depending on its fracture toughness. In the second sense (of the second definition i.e. solid easily broken) a more brittle solid would thus have a lower fracture toughness.

Even if fundamentally different, these two definitions can have things in common. For example, plastic deformation might help a solid to hindered a crack propagation, making it apparently more tough or resistant. It is also interesting that even authors who rely on the first definition sometimes refers also to the second sense. For Example, Hillerborg, who defined the characteristic length presented above, stated in another article that (Hillerborg, 1985) : “If concrete were a perfectly brittle material any small crack or flaw in a region with tensile stresses would cause a running crack, which might lead to a catastrophical failure.” This would only be true if a perfectly brittle solid is considered: 1) to be a material that can be treated with linear fracture mechanics (i.e. no plastic deformation) and 2) to have a fracture toughness equal to zero. This hence is a mix of the two definitions.

The consequence of brittleness being not always a well-defined concept is that a lot of different indices have been proposed to quantify it. In this paper we will focus on those used in the case of pharmaceutical tablets. Three main indices can be found in the literature that are supposed to quantify the brittleness of pharmaceutical tablets.

The first one was proposed by Hiestand *et al.* and is called the Brittle fracture index (BFI)(Hiestand et al., 1977, Hiestand and Smith, 1984). It is defined as follows:

Where σ_{t} is the tensile strength of a normal tablet and σ_{th} is the apparent tensile strength of a tablet with the same characteristics than the previous one but with a hole inserted in the middle.

An extension of the BFI was recently proposed as it was demonstrated that the value of the BFI was influenced by the size of the hole (Croquelois et al., 2020, 2017). This extension was based on the use of a failure criterion called “average-stress criterion”(Whitney and Nuismer, 1974) and made it possible to define a characteristic length a_{0} which can be used to quantify brittleness (i.e. the larger a_{0}, the more brittle). This length can be defined as using the following equation:

where a is the radius of the hole at the center of the tablet, x is the distance to the hole center and f is the stress distribution at hole edge. If a_{0} was previously determined using different hole sizes to lower the uncertainties, it can also be calculated with a single hole size using equation 5.

The second index, called the Brittle/Ductile index (BDI), was introduced by Sönnergard and is defined as (Sonnergaard, 2013):

Where WOF is the work of failure of a tablet during the diametral compression test, F is the failure force during the same test and D is the original diameter of the tested tablet. Note that in a recent publication, Sonnergaard presented also closely related indices but showing that the BDI was the most efficient (Sonnergaard, 2023). We will thus focus on the BDI.

Finally, the more recent index was proposed by Gong and Sun (Gong and Sun, 2015). It is called the tablet brittleness index (TBI) and is defined as:

Where ε_{F} is the elastic strain at fracture during the diametral compression test.

As it can be seen from their definition, these index are based on very different premises, and they can give contradictory results. The aim of the present article is to try to understand, from a fundamental point of view, what exactly do these indices measure and how is this related to the definitions of brittleness presented above. For this, the reasoning will rely on the equations of the mechanics but also on the use of numerical modelling especially in the case of Hiestand’s BFI. An experimental case study will also be presented to illustrate the findings.

### Powders used

Three classical pharmaceutical excipients were used: Microcristalline cellulose (MCC)(Vivapur 200, JRS Pharma, Rosenberg, Germany), Lactose Monohydrate (LAC) (SuperTab 30GR, DFE Pharma, Goch, Germany) and starch (StarTab, Colorcon, West Point, USA). Magnesium stearate (MgSt) (Ligamed MF-2-V, Peter Greven, Bad Münstereifel, Germany) was used for external lubrication.

Read more

Vincent Mazel, Pierre Tchoreloff, Indices for the brittleness of pharmaceutical tablets: A reassessment, International Journal of Pharmaceutics, Volume 645, 2023, 123364, ISSN 0378-5173,

https://doi.org/10.1016/j.ijpharm.2023.123364.