Gradient elasticity with surface energy: mode-I crack problem

Summarization: The solution of the mode-I crack problem is given by using an anisotropic strain-gradient elasticity theory with surface energy, extending previous results by Vardoulakis and co-workers, as well as by Aifantis and co-workers. The solution of the problem is derived by applying the Fourier transform technique and the theory of dual integral and Fredholm integral equations. Asymptotic analysis of the solution close to the tip gives a cusping crack with zero slope of the crack displacement at the crack tip. Cusping of the crack tips is caused by the action of “cohesive” double forces behind and very close to the tips, that tend to bring the two opposite crack lips in close contact. Consideration of Griffith's energy balance approach leads to the formulation of a fracture criterion that predicts a linear dependence of the specific fracture surface energy on increment of crack propagation for such crack length increments that are comparable with the characteristic size of material's microstructure. This important theoretical result agrees with experimental measurements of the fracture energy dissipation rate during fracturing of polycrystalline, polyphase materials such as rocks and ceramics. The potential of the theory to interpret the size effect, i.e. the dependence of fracture toughness of the material on the size of the crack, is also presented. Also, the theory predicts an inverse first power relation between the tensile strength and the size of the pre-existing crack which is in accordance with experimental evidence. Furthermore, it is shown that the effect of the volumetric strain-gradient term is to shield the applied loads leading to crack stiffening, hence the theory captures the commonly observed phenomenon of high-effective fracture energies of rocks and ceramics; the effect of the surface strain-energy term is to amplify the applied loads leading to crack compliance and essentially captures the development of the “process zone” or microcracking zone around the main crack in a brittle material. Thus, the present anisotropic gradient elasticity theory with surface energy provides an effective tool for understanding phenomenologically main crack-microdefect interaction phenomena in brittle materials.Presented on: International Journal of Solids and Structure