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On the Integro-Differential Equation Associated With Diffusive Crack Growth Theory



Y A. Antipov, Tze J. Chuang, H Gao


At high temperatures, polycrystalline materials often suffer creep fracture under prolonged loading conditions. Microstructural examinations revealthat 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 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 derivethe 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 behaviors 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 crack-growth rate for a given applied stress. The stress solutions in analytical form have the merit over numerical form, that they will facilitate the future solution scheme when the analysis is extended to tackle crack growth in transient creep stage wherein both stresses and near-tip crack shapes are changing continuously with time.
Quarterly Journal of Mechanics and Applied Mathematics
Part 2


diffusive crack growth, integro-differential equation, Riemann-Hilbert problem


Antipov, Y. , Chuang, T. and Gao, H. (2003), On the Integro-Differential Equation Associated With Diffusive Crack Growth Theory, Quarterly Journal of Mechanics and Applied Mathematics (Accessed December 10, 2023)
Created May 1, 2003, Updated June 2, 2021