Energy approach to brittle fracture in strain gradient modelling

In this paper we exploit some results in the theory of irreversible phenomena to address the study of quasi-static brittle fracture propagation in a 2- dimensional isotropic continuum. The elastic strain energy density of the body has been assumed to be geometrically non-linear and to depend upon the strain gradient. Such generalized continua often arise in the description of micro-structured media. These materials possess an intrinsic length scale, which determines the size of internal boundary layers. In particular, the non-locality conferred by this internal length scale avoids the concentration of deformations, which is usually observed when dealing with local models and which leads to mesh-dependency. A scalar Lagrangian damage field, ranging from zero to one, is introduced to describe the internal state of structural degradation of the material. Standard Lamé and second gradient elastic coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. Numerical solutions of the model are provided in the case of an obliquelynotched rectangular specimen subject to monotonous tensile and shear loading tests and brittle fracture propagation is discussed.