In this paper an analytical expression for the energy release rate has been derived and put in a form suitable for a numerical analysis of an arbitrary 3-D crack configuration. The virtual crack extension method can most conveniently be used for such a derivation. This method was originally developed from finite element considerations and the resulting expressions were, therefore, based on the finite element matrix formulation [1][2][3][4][5]. In this paper the derivation of the energy release rate leads to an expression which is independent of any specific numerical procedure. The formulation is valid for general fracture behavior including nonplanar fracture and shear lips and applies to elastic materials as well as materials following the deformation theory of plasticity. The body force effect is also included. For 3-D fracture problems it is of advantage to use both an average and a local form of the energy release rate and definitions for both forms are suggested. For certain restrictions on the crack geometry it is shown that the energy release rate reduces to the 3-D form of the J-integral.

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A m ( c r a m roc 75 121 295-5Oi (1992) The electrical conductivity of high-purity single-crystal alumina is determined in a temperature range from 400" to 1300°C. By applying an advanced fully guarded threeterminal measurement technique, reliable conductivity measurements are performed to as low as R-'*cm-'. Gas and surface conduction are measured separately and shown to be negligible. High-purity sapphire exhibits a conductivity of W' ocm-' at 400"C, two characteristic activation energies of 0.4 and 4.8 eV with increasing temperature, and a conductivity of 3 X lo-' W'-cm-' at 1300°C. The fraction of the current carried by ions is determined by electron probe analysis of the electrodes following a 640-h transference test at 1200°C with 4 kV/cm field applied. Only 0.3% of the current at 1200°C is carried by ions. A mathematical model of electrical conduction in sapphire is developed which describes sapphire as a wide-bandgap semiconductor, doped with one dominant donor and one dominant acceptor. The observed conductivity is well described by the model over the entire temperature range from 400" to 1300°C. [