© 2003 by Oxford University Press
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On the Integro-Differential Equation Associated with Diffusive Crack Growth Theory
( 1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA 2 Ceramics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 3 Max Planck Institute for Metals Research, Seestrasse 92, Stuttgart 70174, Germany )
At high temperatures, polycrystalline materials often suffer creep fracture under prolonged loading conditions. Microstructural examinations reveal that nucleation, propagation and linkage of interfacial cracks normal to the principal stress directions are responsible for the premature failure. To simulate service conditions, a semi-infinite crack is considered to grow, in steady state, along a grain boundary via a coupled process of surface and grain-boundary diffusion within an elastic bi-crystal subjected to a remote constant applied stress. Governing equations based on equilibrium and Hooke's law obeyed within the adjoining grains, and matter conservation and Fick's diffusion laws prevailing at both crack surfaces and the interface are employed to derive the singular integro-differential equation for the normal stress distribution along the interface ahead of the moving crack tip.
Using the Mellin transformation, the integral equation is first converted to a functional-difference equation (a Carleman boundary-value problem), and then solved analytically via an approach based on the theory of the Riemann–Hilbert problem on a curve. Asymptotic behaviours of the stress solutions at both ends (that is, near the crack tip as well as in the far field) are provided. Excellent agreement is reached when the full analytical solutions are compared with the existing numerical solutions.
The stress solutions permit the far-field loading intensity to be connected with the boundary conditions containing the parameter of crack velocity at the crack tip, thereby making it possible to predict the crack-growth rate for a given applied stress. The stress solutions in analytical form have the merit, over the numerical form, that they will facilitate the future solution scheme when the analysis is extended to tackle crack growth in the transient creep stage wherein both stresses and near-tip crack shapes are changing continuously with time.
Received 4 November 2002.