The dominant and higher-order asymptotic stress and displacement fields surrounding a stationary crack embedded in a ductile functionally graded material subjected to antiplane shear loading are derived. The plastic material gradient is assumed to be in the radial direction only and elastic effects are neglected. As in the elastic case, the leading (most singular) term in the asymptotic expansion is the same in the graded material as in the homogeneous one with the properties evaluated at the crack tip location. Assuming a power law for the plastic strains and another power law for the material spatial gradient, we derive the next term in the asymptotic expansion for the near-tip fields. The second term in the series may or may not differ from that of the homogeneous case depending on the particular material property variation. This result is a consequence of the interaction between the plasticity effects associated with a loading dependent length scale (the plastic zone size) and the inhomogeneity effects, which are also characterized by a separate length scale (the property gradient variation).
Asymptotic Analysis of a Mode III Stationary Crack in a Ductile Functionally Graded Material
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Kubair, D. V., Geubelle, P. H., and Lambros, J. (October 20, 2004). "Asymptotic Analysis of a Mode III Stationary Crack in a Ductile Functionally Graded Material." ASME. J. Appl. Mech. July 2005; 72(4): 461–467. https://doi.org/10.1115/1.1876434
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