Healing of penny-shaped crack in material with functional gradient structure

Abstract

Composites with a continuously varying volume fraction are called functionally graded materials. As part of research on functionally graded materials, the influence of a continuous axial gradient in elastic properties and a thin, ductile repair layer on a penny-shaped crack is being examined. The body is modeled as an isotropic solid whose shear modulus varies along an axis perpendicular to the crack according to a power-law relationship, while Poisson’s ratio is fixed. The solidified injected filler occupies the crack cavity as a thin layer of variable thickness and transmits tensile forces between the crack surfaces in accordance with the Winkler-type contact reaction. Hankel transforms applied together with the interlayer-modified condition on the crack faces reduce the problem to a Fredholm integral equation of the second kind for the crack-opening displacement. The equation is solved numerically by a quadrature method, and the normalized stress intensity factor is evaluated as a function of the grading parameter, the filler stiffness, and the relative interlayer thickness. The analysis shows that the solution depends on the filler only through an effective stiffness combining these two parameters, so filler stiffness and interlayer thinness are interchangeable: the thinner the interlayer, the more compliant the filler that suffices for the same healing effect. Numerical results show that the recovery effect is sharply nonlinear: even a very ductile filler eliminates most of the stress concentration near the crack perimeter, whereas a further increase in the filler’s stiffness or a reduction in the interlayer thickness yields an increasingly smaller effect. For very thin interlayers, the residual stress concentration decreases to a small fraction of the level characteristic of an open crack. Material grading increases the stress intensity level in an unhealed crack; however, the relative effectiveness of healing increases as the degree of grading increases: when using the same filler, the residual SIF in a body with a strong gradation decreases to a smaller fraction of its value in the ungraded state than in a homogeneous body; therefore, grouting by injection proves to be even more effective in functionally graded materials. The results obtained confirm the possibility of strengthening, through injection, cracked structural elements made of continuously heterogeneous materials.