Singularity-free defect mechanics for polar media

Abstract

We present singularity-free solution for cracks within polar media in which material points possess both position and orientation. The plane strain problem is addressed in this study for which the generalized continua including micropolar, nonlocal micropolar, and gradient micropolar elasticity theories are employed. For the first time, the variationally consistent boundary conditions are derived for gradient micropolar elasticity. Moreover, having reviewed the solution to line defects including glide edge dislocation and wedge disclination from the literature, the fields of a climb edge dislocation within micropolar, nonlocal micropolar and gradient micropolar elasticity are derived. This completes the collection of line defects needed for an inplane strain analysis. Afterward, as the main contribution, using both types of line defects (i.e., dislocation being displacement discontinuity and disclination being rotational discontinuity), the well-established dislocation-based fracture mechanics is systematically generalized to the dislocation-and disclination-based fracture mechanics of polar media for which we have three translations together with three rotations as degrees of freedom. Due to the application of the line defects, incompatible elasticity is employed throughout the paper. Cracks under three possible loadings including stress and couple stress components are analyzed, and the corresponding line defect densities and stress and couple stress fields are reported. The singular fields are obtained once using the micropolar elasticity, while nonlocal micropolar, and gradient micropolar elasticity theories give rise to singularity-free fracture mechanics. IntroductionGeneralized (or extended) continua include (strong) nonlocal theories as well as higher-order (or microcontinuum) and higher-grade (or weak nonlocal) extensions to the classical continuum theories. Within nonlocal elasticity, nonlocal stresses at a material point is expressed as a function of weighted values of the entire stress field. On the other hand, by increasing the order of the theory, microcontinuum field theories (e.g., micromorphic elasticity, microstretch elasticity, micropolar elasticity, and dilatation elasticity) are obtained. Further, higher-grade versions include gradient elasticity theories.A microcontinuum is a continuous collection of deformable point particles [14] for which the stress measures include force stress as well as couple stress [13,54,55]. Microcontinuum field theories address media for which the microstructure is no longer rigid (e.g., polar media). These theories are in fact a two-level continuum models where macroscopic deformation field is enhanced with the microscopic deformation. Within Communicated by Victor Eremeyev, Holm Altenbach.