Page 51 - Fiber Fracture

36 M. Elices and J. Llorca

Table 1. Numerical and experimental results for tensile strength of copper (100)
Method amax @Pa) (100) References
Orowan-Polanyi (Eq. 1) 25 Kelly and Macmillan (1986)
Non-self-consistent KKR 55 Esposito et al. (1980)
Self-consistent ASW, 32 Esposito et al. (1980)
Non-empirical potential 36 Esposito et al. (1980)
Johnson et al. potential 41 Johnson and Wilson (1972)
Morse-function lattice model 7 Milstein and Farber (1980)
Experimental values (whisker) I .7 Brenner (1956)
Experimental values (whisker) 1.7 Kobayashi and Hiki (1973)
Flat crystal under shock wave 17 McQueen and Marsh (1962)

(1981) are: 10.4 GPa in (1 lo), 12.6 GPa in ( 11 1) and 15.2 GPa in (100). Tensile strengths
at 10.9, 22.3 and 15.3 GPa for A1203 whiskers grown in the (Owl), (1 120), and (1100)
crystal directions were measured by Soltis (1965) and estimates of 98,87 and 87 GPa for
the ideal uniaxial tensile strength parallel to (OOOl), (1 120) and (Ioio), respectively, were
computed from the third-order elastic stiffnesses by Gieske (1 968). The Orowan-Polanyi
equation gives a value of 46 GPa for the ideal (0001) tensile strength.
Metals exhibit the maximum stress only in whisker form because they permit
dislocation glide at low stresses, and whiskers are almost free of dislocations. In bcc
metals, improved potential models will lead to a better understanding of the ideal
strength than has so far been gained from either the Orowan-Polanyi approach or the
use of the Morse potential; predictions, for example, of the fracture stress for a-Fe
whiskers in the (1 1 1) direction using the Orowan-Polanyi equation are 46 GPa, where
the maximum tensile stress obtained by Brenner was 13.1 GPa (Brenner, 1956), at an
elongation close to 0.05 (see paper by Kunzi, this volume).
For fcc metal fibres and whiskers, the Orowan-Polanyi equation and first principle
band theory approaches give estimates of the ideal uniaxial tensile stress up to an order
of magnitude above the highest stresses measured (see, for example, Table 1). For
‘perfect’ Cu fibres pulled in tension in the direction (IOO), for example, predictions
of the theoretical tensile stress using the Orowan-Polanyi equation bring 25 GPa. Ab
initio calculations (Esposito et al., 1980), based on non-self-consistent KKR using the
muffin-tin approximation or using the self-consistent augmented spherical wave (ASW)
method, provided values of 55 and 32 GPa, respectively. Other methods, based on a
non-empirical pair potential that can be expressed in terms of the cohesive energy and
can be evaluated when the energy is a function of the nearest-neighbour distance, givc
values of 36 GPa (Esposito et al., 1980) and 41 GPa (Johnson and Wilson, 1972).
Other computations based upon a simple Morse-function lattice model give values of
7 GPa (Milstein and Farber, 1980). Experimental results on whiskers show values of
1.7 GPa (Brenner, 1956; Kobayashi and Hiki, 1973). Brenner noted that Cu did not
cleave but sheared apart, and Kobayashi and Hiki suspected that surface defects were
present. McQueen and Marsh (1962) reported a tensile strength of 17 GPa, but Esposito
et al. (1980) noticed that this value, based on shock-wave intensity, involves techniques
whose validity is difficult to assess.