## Flexure testing beam bending

In flexure testing, a rectangular beam of small thickness and width in comparison with its length is subjected to transverse loads and its central deflection caused by bending is measured. The beam may be supported and loaded in one of two ways (Fig. 2). If the beam is supported at two points and is loaded at two points it undergoes what is known as four-point bending while if it is supported at two points but is loaded at one point it undergoes what is known as three-point bending. In both systems the beam must be supported symmetrically with an overhang at both ends and the loading points must be symmetrical about the mid-point of the specimen. Equations for the calculation of Young's modulus from the applied load F and the deflection of the mid-point of the beam £ can easily be derived, e.g. for four-point bending (Church, 1984): Rg. 2 Geometries for (a) four-point and (b) three-point beam bending. F, applied load; h, beam thickness; b, beam width; I and a, distances between loading points as shown.

for three-point beam bending (Roberts etal., 1991):

4 ttfb

where h and b are the height (thickness) and width (breadth) respectively and / and a are as given in Fig. 2. In both cases the value for Young's modulus for the specimen under test can either be calculated from a single point determination or more commonly from the slope of total load (F) versus central deflection (£).

The four-point beam bending test was first used for pharmaceutical materials by Church (1984) and has since been adapted by Mashadi and

Newton (1987a,b), Bassam etal. (1988, 1990, 1991), Roberts etal. (1989b) and Roberts (1991). Generally beams of varying height have been used -100 mm long by 10 mm wide or 60 mm long by 7 mm wide - and are prepared using specially designed punches and dies (Plate 1). Beams are generally prepared at varying compression pressures to achieve specimens of varying porosity. Although the height of the beam (at constant porosity) is an experimental variable, Bassam etal. (1990) has shown that it does not have a significant effect on the measured modulus.

The testing rig used by these workers consists essentially of three parts (Plate 2): an upper frame (A) with a platform on which the beam is supported on the two outer contact points and which also holds a displacement transducer kept in contact with the lower surface of the beam by means of an elastic band, a block (B) containing the inner contact points resting on the upper surface of the beam, and a lower frame (C) located in the centre of block (B) by means of a ball bearing. In all cases the contact points are cylindrical and mounted in ball-bearing races to allow free movement. The two frames are attached to the moving platens of a tensile-testing machine which allows measurements to be made at varying loading rates. Generally, however, low rates are used (= 1 mm min~') but Bassam etal. (1990) have shown independence of loading rate up to rates of 15 mm min-1. Deflection (jim)

Hg. 3 A typical load-deflection curve for a beam 100 mm long of microcrystalline cellulose.

Deflection (jim)

Hg. 3 A typical load-deflection curve for a beam 100 mm long of microcrystalline cellulose.

A typical load-deflection curve for a beam 100 mm long of microcrystal-line cellulose compacted to a porosity of 14.6% is shown in Fig. 3. Generally, each beam is tested five times and for the specimen shown in Fig. 3 calculations of the Young's modulus of elasticity gave a value of 3.53 ± 0.07 GPa, i.e. a coefficient of variation of 2%; such reproducibility is not uncommon with this method of measurement.

A disadvantage of the four-point beam test method is that it invariably requires large specimens and hence large quantities of materials (15-20 g). In addition, high-tonnage presses are needed to prepare the specimens thus exacerbating problems with cracking and lamination on ejection from the die. It was to overcome these difficulties that Roberts etal. (1989b) developed a three-point beam testing method that uses beams prepared from 200 mg of material. The beams in this case are 20 mm long by 7 mm wide and are stressed by applying a static load of 0.3 N with an additional

3-potrri bend rig

Measure probe linear motor coil

fig. 4 Schematic diagram of the thermal mechanical analyser used to measure Young's modulus.

dynamic load of ±0.25 N (at a frequency of 0.17 Hz) using a thermal mechanical analyser (Mettler Instruments TMA40). The schematic diagram (Fig. 4) shows the position of the linear differential displacement transducer attached to the measuring probe and the modification made to enable the application of extra loading while Plate 3 shows the three-point bending rig with the measuring probe. In operation a calibration run is first performed to eliminate distortions in the sensor and other parts of the displacement measuring system and then 20 measurements of specimen displacement are undertaken to an accuracy of ±0.005 /xm. The Young's modulus of elasticity of the specimen is then calculated from the mean displacement corrected for distortions using equation 8, where in this case F is the applied dynamic load. Extensive testing by Roberts etal. (1989b) and Roberts (1991) has shown equivalence between this test and the conventional four-point beam test despite the fact that it is generally accepted that in the latter there is a more uniform stress pattern over the central section of the beam and little contribution from shear stresses.

A problem associated with the analysis of data for specimens prepared from particulate solids is in the separation of the material property from that of the specimen property which by definition includes a contribution by the porosity of the specimen. All workers have found that for all materials there is a decrease in Young's modulus with increasing porosity (Fig. 5). Numerous equations have been published which describe this relationship (Dean and Lafez, 1983). Certain equations are based on theoretical considerations (Wang, 1984; Kendall etal., 1987) while others are empirical curve-fitting functions (Spriggs, 1961; Spinner etal., 1963). Recently, Bassam etal. (1990) have reported a comparison of all the equations currently in use for data generated from four-point beam bending on 15 pharmaceutical powders ranging from celluloses and sugars to inorganic materials such as calcium carbonate and have concluded that the best overall relationship is the modified two-order polynomial (Spinner etal., 1963)

where Ea is the Young's modulus at zero porosity and E is the measured modulus of the beams compacted at porosity, P; fx and /2 are constants. However such a conclusion may not be universal as during extensive studies on a wider range of materials including drugs, Roberts (1991) concluded that an exponential relationship (Spriggs, 1961) is the preferred option for data generated using the three-point beam testing method, i.e.

where b is a constant. It is interesting to note that, on average, the extrapolated values of Ea (Young's modulus at zero porosity) calculated using Fig. 5 The effect of porosity on the measured Young's modulus of PTFE; testosterone propionate; ♦, theophylline (anhydrous). Data generated using three-point beams (taken from

Porosity

Fig. 5 The effect of porosity on the measured Young's modulus of PTFE; testosterone propionate; ♦, theophylline (anhydrous). Data generated using three-point beams (taken from

both equations are only marginally different (Bassam etal., 1988,1990).

The analysis of Young's modulus at zero porosity thus provides a means of quantifying and categorizing the elastic properties of powdered materials. Table 1 shows literature data for a variety of pharmaceutical excipients and drugs determined using beam bending methods. It can be seen that the values vary over two orders of magnitude ranging from hard rigid materials with very high moduli (e.g., the inorganics) to soft elastic materials with low moduli (e.g., the polymeric materials). As a result a rank order of increasing rigidity of tabletting excipients can be listed: starch < microcrystalline celluloses < sugars < inorganic fillers with variations in the groups dependent on chemical structure as well as the preparation and pretreatment routes (including particle size).

The effects of particle size are distinguishable within the celluloses and a-lactose monohydrate. In the former there is a small increase with decreasing particle size while in the latter the increase is much greater. Whereas in the former the effect is probably due to an increase in contact area, in the latter the effect is due to specimen defects in that the specimens used by Bassam etal. (1990) contained microscopic flaws and cracks. Recent work by Roberts (1991) has shown that it is necessary to eliminate all specimens with cracks otherwise the extrapolated modulus values to zero porosity are inconsistent.

For the lactose samples the rank order of increasing rigidity is spray dried < /3-anhydrous < a-monohydrate, consistent with the findings of workers describing the compaction properties of the materials using instrumented tabletting machines (Fell and Newton, 1971; Vromans etal. 1986). The variations in the cellulose samples can be attributed to subtle differences in the manufacturing preparative technique. However, it is known that this factor can also affect the equilibrium moisture content of the samples with Unimac samples attaining a lower equilibrium moisture content than Avicel samples. As it is known that increasing moisture content can lead to a decrease in Young's modulus for microcrystalline cellulose (Bassam etal., 1990) the differences in modulus between sources listed in Table 1 would be expected to increase if all materials were compared at equivalent moisture contents.

All the drugs tested exhibited low moduli equivalent to the polymeric materials. This is not totally unexpected as it is known that many organic solids including drugs form glasses which exhibit anomalous endotherms that resemble glass transitions and can therefore be regarded as possessing a certain amount of mobility.

Recently, Roberts etal. (1991) have investigated the relationship between the Young's modulus of a variety of drugs and excipients using three-point beam bending and their molecular structure based on intermolecular interactions using the concept of cohesive energy density (CED). They found a direct relationship of the form

where CED is expressed in units of MPa and Young's modulus in units of GPa. This equation compares favourably with that derived from the Tobolsky (1962) equation relating the bulk modulus, K, of a face-centred cubic lattice at 0° K to the cohesive energy density:

Since K is related to E by the equation:

where u is the Poisson's ratio (taken at 0.3 for the majority of pharmaceutical materials) then combining equations 12 and 13 gives:

Table 1 Young's modulus at zero porosity measured by flexure testing

Young's modulus

Table 1 Young's modulus at zero porosity measured by flexure testing

Young's modulus

 Material Method (GPa) Celluloses Avicel PH101 4PB (100 X 10) EXP 10.3 Avicel PH101 4PB (100 x 10) EXP 9.7 Avicel PH101 4PB (100 x 10) POLY 9.2 Avicel PHI01 4PB (100 x 10) EXP 9.0 Avicel PH101 3PB (20 x 7) EXP 7.8 Avicel PHI01 4PB (100 x 10) EXP 7.6 Avicel PHI01 4PB (60 x 7) EXP 7.4 Avicel PH102 4PB (100 x 10) POLY 8.7 Avicel PH102 4PB (100 x 10) EXP 8.2 Avicel PH105 4PB (100 x 10) EXP 10.1 Avicel PHI05 4PB (100 x 10) POLY 9.4 Emcocel 4PB (100 X 10) EXP 9.0 Emcocel 4PB (100 x 10) POLY 7.1 Emcocel (90M) 4PB (100 x 10) EXP 9.4 Emcocel (90M) 4PB (100 x 10) POLY 8.9 Unimac (MG100) 4PB (100 X 10) EXP 8.8 Unimac (MG100) 4PB (100 x 10) POLY 8.0 Unimac (MG200) 4PB (100 x 10) EXP 8.0 Unimac (MG200) 4PB (100 x 10) POLY 7.3 Elcema {PI00) 4PB (100 x 10) EXP 8.6 Sugars Sorbitol instant 4PB (100 x 10) EXP 45.0 a-Lactose monohydrate 3PB (20 x 7) EXP 24.1 a-Lactose monohydrate 4PB (100 x 10) POLY 3.2 Lactose /3 anhydrous 4PB (100 x 10) POLY 17.9 Lactose 0 anhydrous 4PB (100 X 10) POLY 18.5

50 Roberts and Rowe (1987c)

50 Bassam etal. (1990)

50 Bassam etal. (1988)

50 Roberts et al. (1989b)

50 Roberts etal. (1989b)

50 Roberts etal. (1989b)

90 Bassam etal. (1990)

90 Bassam etal. (1988)

20 Bassam etal. (1988)

20 Bassam etal. (1990)

56 Bassam etal. (1988)

56 Bassam etal. (1990)

90 Bassam etal. (1988)

90 Bassam etal. (1990)

38 Bassam etal. (1990)

103 Bassam etal. (1988)

103 Bassam etal. (1990)

Roberts and Rowe (1987c)

Mashadi and Newton (1987a) 20 Roberts etal. (1991)

63 Bassam etal. (1990)

149 Bassam etal. (1990)

149 Bassam et al. (1991)

 Lactose (spray dried) 4PB (100 x 10) EXP 13.5 - Roberts and Rowe (1987c) Lactose (spray dried) 4PB (100 x 10) POLY 11.4 125 Bassam etal. (1990) Dipac sugar 4PB (100 x 10) POLY 13.4 258 Bassam etal. (1990) Mannitol 4PB (100 x 10) POLY 12.2 88 Bassam etal. (1990) Polysaccharides Starch 1500 4PB (100 x 10) EXP 6.1 - Roberts and Rowe (1987c) Maize starch 4PB (100 x 10) POLY 3.7 16 Bassam etal. (1990) Inorganics Emcompress 4PB (100 x 10) EXP 181.5 - Roberts and Rowe (1987c) Calcium carbonate 4PB (100 X 10) POLY 88.3 8 Bassam etal. (1990) Calcium phosphate 4PB (100 x 10) POLY 47.8 10 Bassam etal. (1990) Polymers PVC 4PB (100 x 10) POLY 4.4 - Bassam etal. (1991) PVC 3PB (20 x 7) EXP 4.1 - Roberts etal. (1991) Stearic acid 3PB (20 X 7) EXP 3.8 62 Roberts etal. (1991) PTFE 4PB (100 x 10) EXP 0.81 - Roberts etal. (1989b) PTFE 4PB (100 x 10) POLY 0.71 - Bassam etal. (1991) PTFE 3PB (20 X 7) EXP 0.71 - Roberts etal. (1989b) Drugs Theophylline (anhydrous) 3PB (20 X 7) EXP 12.9 31 Roberts etal. (1991) Paracetamol DC 3PB (20 x 7) EXP 11.7 120 Roberts (1991) Caffeine (anhydrous) 3PB (20 x 7) EXP 8.7 38 Roberts etal. (1991) Sulphadiazine 3PB (20 x 7) EXP 7.7 9 Roberts etal. (1991) Aspirin 3PB (20 X 7) EXP 7.5 32 Roberts etal. (1991) Ibuprofen 3PB (20 X 7) EXP 5.0 47 Roberts etal. (1991) Phenylbutazone 3PB (20 x 7) EXP 3.3 50 Roberts etal. (1991) Testosterone propionate 3PB (20 X 7) EXP 3.2 85 Roberts etal. (1991)

PTFH, poiytetrafluoroethyiene; PVC, polyvinyl chloride; 4PB = four-point beam; 3PB = three-point beam; EXP = equation 10; POLY = equation 9; the beam dimensions (length x width in mm) are given in parentheses.

PTFH, poiytetrafluoroethyiene; PVC, polyvinyl chloride; 4PB = four-point beam; 3PB = three-point beam; EXP = equation 10; POLY = equation 9; the beam dimensions (length x width in mm) are given in parentheses.

It can be seen that the moduli predicted from this equation are somewhat higher than those measured (Fig. 6). This is thought to be due to the fact that all the measurements were carried out at 298° K and hence the measured modulus would be lower. The significance of this finding is in the recognition of the validity of both the test method and the data manipulation (i.e., the extrapolation to zero porosity).