The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses

An investigation of asymptotic crack tip singular fields and their domain of validity is carried out for mode I cracks in solids characterized by the phenomenological strain gradient plasticity theory proposed by Fleck NA, Hutchinson JW. (Strain gradient plasticity. In: Hutchinson JW, Wu TY, editors. Advances in applied mechanics, vol. 33. New York: Academic Press, 1997. pp. 295–361.) Separable near-tip singular fields are determined where fields quantities depend on the radial and circumferential coordinates (r,θ) according to rpf θ . The singular field is completely dominated by the strain gradient contributions to the constitutive law. In addition to the asymptotic analysis, full field numerical solutions are obtained by a finite element method using elements especially suited to the higher order theory. It is found that the singular field provides a numerically accurate representation of the full field solution only within a distance from the tip that is a tiny fraction of the constitutive length parameter. The constitutive theory itself is not expected to be valid in this domain. Curiously, the normal traction acting across the extended crack line ahead of the crack tip is found to be compressive in the singular field. The conclusion which must be drawn is that the singular field has a tiny domain of mathematical validity (neglecting crack face interaction), but no domain of physical validity. The significant elevation of tractions ahead of the crack tip due to strain gradient hardening occurs at distances from the crack tip which are well outside this tiny domain in a region where the plasticity theory is expected to be applicable. The asymptotic singular fields are incapable of capturing the effect of traction elevation.