A graded elastic modulus concept to eliminate stress or strain energy density singularity at sharp notches and cracks, with consequent elimination of size-scale effect on strength

Abstract

It has been recently suggested by the author that in the classical problem of a sharp wedge or crack loaded in plane (mode I and/or mode II), the stress singularity can be removed by grading the elastic properties of the underlying material from the notch tip by using a power law, $E\sim r^{\beta }$. While the treatment is extended to the case of mode III (antiplane shear) which permits closed form results, we also discuss two ways to deal with the likely effect of material’s grading on strength. In one, already explored in the previous paper, the strength is a power law of the modulus, and we suggest an “optimal” design by keeping the dominant stress constantly equal to the strength. In a second method, we propose to cancel the singularity in the strain energy density, which requires a much stronger grading, and we also possibly take into account that the critical strain energy density is a power law of the modulus. Noticing that only in the presence of a singularity a length scale can be defined experimentally by testing a very large notch and a very small one, according to the Theory of Critical Distances (TCD), the effect of cancelling singularity also implies independence on size/scale and constant strength. It is concluded that the technique is much more powerful than drilling a hole or rounding the tip of the notch/crack. Moreover, if a “smart” material could be designed to damage itself as to reduce its modulus when near a high stress concentration according to our prescriptions, it would naturally self-heal, opening up interesting applications.