## Single edge notched beam

In this test a prenotched rectangular beam of small thickness and width in comparison to its length is subjected to transverse loads and the load at fracture measured. As with the beams used for the determination of Young's modulus, loading can be by either three or four points (Fig. 14). The single edge notched beam test has been the subject of much research leading to the specification of standard criteria for test piece geometry (Brown and Srawley, 1966; British Standards Institution, 1977) dependent on the material to be studied, e.g.: Fig. 14 Geometries for (a) four-point and (b) three-point single edge notched beam. F, applied load; h, beam thickness; b, beam width; I and a, distances between loading points as shown;

c, crack length.

Fig. 14 Geometries for (a) four-point and (b) three-point single edge notched beam. F, applied load; h, beam thickness; b, beam width; I and a, distances between loading points as shown;

c, crack length.

To take account of the uncertainties such as underestimation of the K,c value and the possibility of the test not meeting some of the other validity criteria set down in British Standard 5447 (1977) it is recommended that

larger pieces be used where c and h should be at least 4

In addi tion, Brown and Srawley (1966) have recommended that:

rK V

In all cases ay is the yield stress of the material under test.

Equations for the calculation of the critical stress intensity factor from the applied load, F, and geometry of the beam can be derived, e.g. for four-point beam bending (Mashadi and Newton, 1987a,b):

2 bh2

and for three-point beam bending (Roberts etal., 1993):

IbhH

where 7 is a function of the specimen geometry expressed as a polynomial of the parameter c/h:

"Y = A0 +

r c

2 „ fc|

3

+ A2\-

+ i43 7

+ A*

ä

IK

IK

where the coefficients have the values shown in Table 10. These equations assume that the artificially induced crack which becomes unstable has zero width, extends the full width/breadth of the specimen and has a depth that is precisely known. Furthermore, the cross-section of the specimen and of the notch must be of sufficient size relative to the microstructural features that the observed response to loading is representative of the bulk material.

The four-point single edge notched beam was first used for pharmaceutical materials by Mashadi and Newton (1987a,b) and has since been adopted by York etal. (1990). In all cases large rectangular beams 100mm long x 10 mm wide of varying height are prepared using the same punches and dies as those used to prepare specimens for the determination of Young's modulus. Notches of varying dimensions and profiles have been

 Type of loading a, a2 a4 Four point + 1.99 -2.47 +12.97 — 23.17 + 24.80 Three point l=* + 1.96 -2.75 + 13.66 — 23.98 + 25.22 + 1.93 -3.07 + 14.53 -25.11 + 25.80

introduced by cutting either using a simple glass cutter (Mashadi and Newton, 1987a,b) or a cutting tool fitted into a lathe (York etal., 1990). The latter method has allowed notches of different profiles and dimensions to be accurately cut and investigated. While the arrowhead type notch did appear to influence the measured value of KIC for beams of microcrystal-line cellulose, the effect was much reduced from straight-through notches and hence the latter were recommended (York etal., 1990).

The load required for failure of the specimens under tension is measured using the same testing rig as that described previously for the determination of Young's modulus. Loading rates of between 0.025 mm min"1 (Mashadi and Newton, 1987a,b) and 100mm min-1 (York etal., 1990) have been used - the latter workers noted a small rise (approximately 10%) in the measured KIC of beams of microcrystalline cellulose for a 100-fold increase in applied loading rate.

As with the measurement of Young's modulus the four-point test requires large beams and consequently large amounts of material. In order to minimize the latter, especially for the measurement of the critical stress intensity factor for drugs under development, Roberts etal. (1993) have developed a three-point test using specimen dimensions and testing rig (Plate 6) similar to that described earlier for Young's modulus although in this case a ten-someter has been used to stress the specimen. Using beams of dimensions 20 mm long X 7 mm wide of varying height with two types of notches (a V notch cut by a razor blade pressed into the surface and a straight-through notch cut by a small saw blade) the authors were able to show equivalence with data generated by York etal. (1990) for the four-point single edge notched beam (Fig. 14).

As can be seen from Fig. 15, specimen porosity has a significant effect on the measured KIC. As porosity decreases KIC increases indicating more resistance to crack propagation. During the initial stages of compression the large pores which control the strength of the specimen are removed first followed by the smaller ones. As the powder becomes more consolidated it becomes less brittle and is able to absorb greater loads before failure. It Fig. 15 The effect of porosity on the measured critical stress intensity factor for: ■, V notch; and straight through notch of Avicel PH101 under three-point loading. X, data from York eta/. (1990) for some material under four-point loading. The line represents equation 40 for all points.

Fig. 15 The effect of porosity on the measured critical stress intensity factor for: ■, V notch; and straight through notch of Avicel PH101 under three-point loading. X, data from York eta/. (1990) for some material under four-point loading. The line represents equation 40 for all points.

is obvious from the results in Fig. 15 that the relationship between KIC and porosity is not linear as suggested by Mashadi and Newton (1987a,1987b, 1988). In this respect York etal. (1990) have investigated the application of both the two-term polynomial equation of the type:

and the exponential equation of the type:

where KICo is the critical stress intensity factor at zero porosity, KIC is the measured critical stress intensity factor of the specimen at porosity P and b, fx, and /2 are constants. (It should be noted that these equations are analogous to those used previously for Young's modulus.) Both the relationships give low standard errors and high correlation coefficients for microcrystalline cellulose from various sources. As with Young's modulus, the exponential relationship is the preferred option for pharmaceutical materials (Roberts etal., 1993).

Table 11 Critical stress intensity factors measured using a single-edge notched beam

Critical stress intensity factor

 Material Method (MPam1 Celluloses Avicel PH101 4PB (100 x 10) LIN 1.21 Avicel PHI02 4PB (100 x 10) EXP 0.76 Avicel PH101 4PB (100 x 10) EXP 0.87 Avicel PH105 4PB (100 x 10) EXP 1.33 Emcocel (90M) 4PB (100 x 10) EXP 0.80 Emcocel 4PB (100 x 10) EXP 0.92 Unimac (MG200) 4PB (100 X 10) EXP 0.67 Unimac (MG100) 4PB (100 x 10) EXP 0.80 Avicel PHI02 4PB (100 x 10) POLY 0.91 Avicel PH101 4PB (100 x 10) POLY 0.99 Avicel PHI05 4PB (100 x 10) POLY 1.42 Emcocel (90M) 4PB (100 x 10) POLY 0.83 Emcocel 4PB (100 x 10) POLY 0.80 Unimac (MG200) 4PB (100 x 10) POLY 0.76 Unimac (MG100) 4PB (100 x 10) POLY 1.05 Avicel PH101 3PB (20 x 7) EXP 0.76

Particle size Gun)

Reference

 50 Mashadi and Newton (1987b) 90 York etal. 1990) 50 York etal. 1990) 20 York etal. 1990) 90 York etal. 1990) 56 York et al. 1990) 103 York et al. 1990) 38 York et al. 1990) 90 York etal. 1990) 50 York etal. 1990) 20 York etal. 1990) 90 York etal. 1990) 56 York etal. 1990) 103 York etal. 1990) 38 York etal. 1990) 50 Roberts eta Lactose /3 anhydrous a-Lactose monohydrate Sucrose Sorbitol instant Drugs Ibuprofen Aspirin Paracetamol DC Paracetamol Others Sodium chloride Adipic acid 4PB = four-point beam; 3PB = three-point beam; EXP = equation 40; POLY dimensions (length and width) are given in parentheses. 149 Roberts et al. (1993) 20 Roberts etal. (1993) 74 Roberts etal. (1993) Mashadi and Newton (1987a) 47 Roberts etal. (1993) 32 Roberts etal. (1993) 120 Roberts etal. (1993) 15 Roberts etal. (1993) 20 Roberts etal. (1993) 176 Roberts etal. (1993) = equation 39; LIN = linear regression; the beam A number of pharmaceutical materials have been measured using both three- and four-point beam bending (Table 11). Of all the excipients tested microcrystalline celluloses exhibited the highest values of KICo; real differences existed between the materials obtained from different sources. In addition, there is also a particle size effect in that, for each of the three sources of material, the critical stress intensity factor increased with decreasing particle size. The sugars exhibit intermediate values with relatively low values for the drugs. It is interesting that anhydrous & lactose has a critical stress intensity factor approximately twice that of a-lactose monohydrate (c.f. indentation hardness measurements in Table 4).