Gradient effects in fracture mechanics for nano-structured materials

•We model the fracture mechanics problem using the gradient elasticity.•We examine smoothness of solution to specify the stresses at the crack tip.•We show dependence of singular solution from the large-scale structure parameter.•We establish the correlation of size characteristic and the length-scale parameter.•We formulated J-integral equation for the adhesion models.

Gradient elastic effects associated with the existence of a cohesion type interphase layer within a simple but robust fracture model are discussed. The classical and gradient solutions whose properties are described by harmonic and Helmholtz equations are compared. It is shown that the use of gradient solutions can explain various fracture mechanisms for microcrystalline structures depending on the length-scale parameter that is defined in terms of standard characteristics of material. Here, we have established the equations for energy tensor in gradient model that takes into account the adhesion interactions in media. Finally, the law of conservation for energy tensor and J-integral equation are formulated within the framework of gradient cohesion–adhesion continuum model.