d fig. 18 Schematic diagram showing cracking around a Vickers indent. C0, crack length; d, length of diagonal of indent; a, length of half-diagonal of indent.
Before describing the technique it is important to realize the assessment of Kc is strictly semiempirical because, first, the analytical solutions of the stress field around indentations have not been solved and only approximate solutions have been derived, and second, the deformation field is not homogeneous and anisotropy and fracture complicate the problem.
The most common solution is an equation based on the model of Evans and Charles (1976) and from experimental observations of Marshall and Lawn (1979):
o where Fis indentation load, Ca is the crack length (as in Fig. 18) and x is a constant which can include terms reflecting the plastoelasticity of the system, e.g. (E/H)m where m is a constant which varies depending on the calibrating materials assuming that Kc = K1C, where K,c is from conventional fracture testing. Obviously, the wider the range of solids used, the more universal is the equation and in this respect Evans and Charles (1976) have used ceramic data from the double torsion test as calibrating data obtaining a value of x of 0.0824.
The most widely used equation for determining Kc is from Antis et al. (1981) using both measured and literature data for H, E and KIC (double cantilever beam test) for ceramics to allow calibration:
where # is a calibration constant equivalent to 0.016.
In a review article Ponton and Rawlings (1989a,b) analysed 19 indentation fracture equations for their ability to equate to conventional testing and found an equation by Lankford (1982) the most universal:
where $ is a calibration constant equivalent to 0.0363 and a is the half diagonal or a = d/2.
Only two papers have examined the indentation fracture test as a means of determining the critical stress intensity factor of pharmaceuticals (Duncan-Hewitt and Weatherly, 1989a,b). In the first of these, Duncan-Hewitt and Weatherly (1989a) evaluated the indentation test using sucrose crystals. Microindentation was performed using a Leitz-Wetzlar Miniload hardness tester (Vickers pyramidal diamond indenter), applying loads of 147 mN, with the indentations and cracks measured using a light microscope (Leitz). Large crystals were prepared (1-4 mm diameter) by slow evaporation of a saturated aqueous solution at 23 °C over a period of 3-6 months. The prismatic crystals were washed in ethanol to remove traces of crystallization solution and were stored under controlled conditions before testing. Furthermore, some specific crystal faces - (100), (010) and (001) - were prepared by either abrading with decreasing grades of emery paper or by cleavage. Crystals were mounted in plasticine prior to testing. The authors found that fractures appeared anisotropic and that if crystals were tested immediately after polishing, fracture was either suppressed or significantly decreased, e.g. crack lengths after 2min, 4min and 20min polishing were 0/tm, 16/tm and 34/tm, respectively.
In addition to using equation 45 of Antis etal. (1981), Duncan-Hewitt and Weatherly (1989a) examined the equation of Laugier (1987):
where # is a calibration constant equivalent to 0.0143 and I = C0 — a.
The authors found that the values of Kc using the two equations were similar (the mean for the various faces are given in Table 13). Furthermore, they reported that equation 47 (Laugier, 1987) appeared to emphasize the apparent fracture anisotropy. The fracture plane with the lowest value was the (100) in agreement with the easiest to cleave plane (although the (101) had the lowest Kc value) and the plane with greatest Kc was the (001) plane. It is interesting to note that the (100) plane had the hardest surface whereas the (001) plane had the softest (Table 6).
Calculations have been performed to analyse the ability of all the fracture indentation equations (see Table 13) to equate to the value of sucrose from flexure testing (Roberts etal., 1993), e.g. the equivalence of Kc to KICo,
Table 13 Critical stress intensity factors for sucrose
Critical stress intensity factor (MPam1/2)
Evans and Charles (1976) Antis etal. (1981) Laugier (1987) Lankford (1982)
Single edge notched beam
Roberts etal. (1993)
and these data are presented in Table 13. In all cases the indentation technique gives values of the critical stress intensity considerably lower than the flexure method, with the value calculated using the Lankford (1982) equation giving the closest agreement and confirming the study by Ponton and Rawlings (1989a,b) using ceramics.
In a later paper Duncan-Hewitt and Weatherly (1989b) published further indentation critical stress intensity factors calculated from the Antis et al. (1981) relationship (equation 45). These are shown in Table 14 with the corresponding data from single edge notched beam (three point) beam testing (Roberts etal., 1993). Although the rank order is the same the differences in magnitude of the values are large (with the exception of sodium chloride). A possible explanation for these differences is that in the indentation test the theory is not exact and the equations are derived from calibration with ceramics, i.e. materials with plastoelastic properties considerably different from pharmaceutical materials.
Despite its shortcomings the indentation technique has certain advantages over other testing methods as it can be used on small samples, e.g. single crystals, specimen preparation is relatively simple and the indentation hardness and Young's modulus can be determined simultaneously.
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