Mode III crack propagation in couple stress elastic materials.

The mechanical behaviour of most materials with microstructure, like composites, cellular materials, foams, masonry, bone tissues, glassy and semicrystalline polymers, is strongly influenced by the microstructural characteristic lengths, especially in the presence of large (or strain) gradients (Lakes, 1986). These findings stimulated the development of generalized theories of continuum mechanics such as micropolar elasticity and indeterminate couple stress elasticity (Koiter, 1964). Due to the effects of the characteristic lengths, these generalized theories predict dispersive wave propagation in microstructured media. Moreover, they predict different asymptotic behaviour for some components of the deformation and stress fields near singular points, compared to the classical elasticity theory. Due to the complexity of the equations of motion provided by the couple stress elastic theory with rotational inertia, only few crack propagation problems have been considered in the literature, most of them have been solved numerically and very few full-field solution have been worked out in closed form. However, a simple asymptotic characterization of the crack tip fields is not sufficient to analyse the fracture behaviour of materials with microstructure. This is due to the fact that the asymptotic solution is valid in a region close to the crack tip which is smaller than the characteristic lengths of the material, and thus it is of scarce physical relevance. A full field analysis is then required in order to grasp the qualitative and quantitative behaviour of the solution in a larger region and to be able to judge on the stress level supported by the material.The full field solution obtained by Radi (2008) for a Mode III stationary crack by using Fourier transforms and Wiener–Hopf technique, shows that ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution. However, this zone is found to have limited physical relevance and to become vanishing small for a characteristic length in torsion of zero. In this limit, the solution recovers the classical KIII field with square root stress singularity. Outside the zone ahead of the crack tip where the total shear stress is negative, the full field solution exhibits a bounded maximum for the total shear stress, whose magnitude was adopted as a measure of the critical stress level for crack advancing. The introduction of a stress based fracture criterion thus defines a critical stress intensity factor increasing with the characteristic length in torsion.In this paper the previous analysis is extended to the case of dynamic Mode III crack propagation, subject to antiplane loading applied on the crack surfaces, in order to study the effects of inertia and crack-tip speed on the stress and deformation fields, as well as the variation of the fracture toughness due to the presence of microstructure. In particular, we assume that the crack tip speed is smaller than the shear wave velocity. The constitutive behaviour is described by the indeterminate theory of couple stress elasticity developed by Koiter. Differently from classical elastic theories, this reasonably simple model is able to account for the underlying microstructure as well as for the strong size effects arising at small scales. The analytical full-field solution is addressed making use of Fourier transform and Wiener–Hopf technique. Closed-form solutions are provided for vanishing or small contribution from rotational inertia. The analysis confirms and extends earlier results on stationary cracks (Radi, 2008) by including the effects of crack velocity and rotational inertia. By adopting the criterion of maximum total shear stress previously introduced, we also discuss the effects of microstructural parameters on the stability of crack propagation. It follows, indeed, that propagation is unstable if the maximum shear stress occurring ahead of the crack tip during steady propagation is increasing as the crack tip speed increases.