Dislocation-based fracture mechanics within nonlocal and gradient elasticity of bi-Helmholtz type – Part I: Antiplane analysis

•Dislocation-based fracture is successfully applied to nonlocal and gradient elasticity of bi-Helmholtz type.•Gradient elasticity of bi-Helmholtz type results in a full nonsingular fracture theory.•Nonlocal elasticity of bi-Helmholtz type results in a nonsingular nonlocal stress field for plane weakened by a crack.•Crack tip plasticity is captured without any ad hoc assumption. In the present paper, the dislocation-based antiplane fracture mechanics is employed for the analysis of Mode III crack within nonlocal and (strain) gradient elasticity of bi-Helmholtz type. These frameworks are appropriate candidates of generalized continua for regularization of classical singularities of defects such as dislocations. Within nonlocal elasticity of bi-Helmholtz type, nonlocal stress is regularized, while the strain field remain singular. Interestingly, gradient elasticity of bi-Helmholtz type (second strain gradient elasticity) eliminates all physical singularities of discrete dislocation including stress and strain fields and dislocation density while the so-called total stress tensor still contains singularity at the dislocation core. Based on the distribution of dislocations, a fracture theory with nonsingular stress field is formulated in these nonlocal and gradient theories. Strain and displacement fields within nonlocal fracture theory are identical to the classical ones. In contrast, gradient elasticity of bi-Helmholtz type leads to a full nonsingular fracture theory in which stress, strain and dislocation density are regularized. However, the singular total stress of a discrete dislocation results in singular total stress of the plane weakened by a crack. Within classical fracture mechanics, Barenblatt’s cohesive fracture theory assumes that cohesive forces is distributed ahead of the crack tip to model crack tip plasticity and remove the stress singularity. Here, considering the dislocations as the carriers of plasticity, the crack tip plasticity is captured without any assumption. Once the crack is modeled by distributing the dislocations along its surface, due to the gradient theory, the distribution function gives rise to a non-zero plastic distortion ahead of the crack. Consequently, regularized solutions of crack are developed incorporating crack tip plasticity.