Asymptotic Fracture Modes in Strain-Gradient Elasticity: Size Effects and Characteristic Lengths for Isotropic Materials

Size-effects characterize the fracture process of many engineering materials. Their modelling calls for material constitutive relations which are not indifferent, as standard elasticity, to variations of scale: strain-gradient elasticity or plasticity have often served the purpose. The three classical crack opening problems of fracture mechanics are here solved within the framework of linear strain-gradient elasticity for the most general isotropic material. Apart from the Lame constants, this is completely identified by five additional moduli, modelling its micro-structural characteristics. This general setting allows for recovering, as particular cases, previous analyses in (Gourgiotis and Geogiadis in J. Mech. Phys. Solids 57(11): 1898-1920, 2009), relative to modes I and II for the so-called Simplified Mindlin materials (17), and in (Radi in Int. J. Solids Struct. 45(10): 3033-3058, 2008), relative to mode III for couple-stress materials. More importantly, having a rather refined material description allows for understanding how the energy release rate is affected by the actual values of the characteristic lengths. Hence we demonstrate the strengthening effect of suitable microstructures and their optimality in order to provide cohesive like actions on the crack lips. Despite being limited to the linear elastic hypothesis, the solutions found constitute an useful insight to develop more complex, possibly nonlinear, constitutive relationships for the strain-gradient terms as for plastic or viscoelastic materials.