Strain Gradient Elasticity for Antiplane Shear Cracks: A Hypersingular Integrodifferential Equation Approach

How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”

“…(see e.g. [43], [44], [45], [46], [47], [48], [49] , [76], [87], [55], [103], [81], [120], [101], [12], [13], [75], [102], [74], [9], [124], [125], [128], [42], [79], [117])). Here, simply for the particular taste of the authors, it is underlined the possibility of introducing N-th order models for describing some bio-mechanical phenomena ( [80], [82], [70], [73], [132]) or those damage phenomena occurring in crack formation and growth (see e.g.…”

“…Also many efforts are directed towards more or less mathematically rigorous homogenization procedures leading to continua which at a "macroscopic" level behave as higher gradient or Cosserat Continua (see e.g. [101], [12], [13], [75], [102], [74], [9], [124], [125], [128], [42], [79], [117]). Homogenization is also involved in the analysis of dissipation effects in many particle systems in the attempt of describing a finite dimensional system (in which length and time scales may be relevant) through a continuum approximation by the Langevin equation in the context of Brownian motion (see e.g.…”

Mode III crack growth in linear hardening materials with strain gradient effects

“…These investigations found that the strength of the stress singularity turns out to be extremely weak for the typically small values of the strain hardening for ductile metals. The effects of elastic strain gradients on the stress concentration at the tip of a mode III stationary crack were considered by Vardoulakis et al (1996), Zhang et al (1998), Fannjiang et al (2002 and Georgiadis (2003). While in Vardoulakis et al (1996) and Fannjiang et al (2002), strain gradients are introduced through the second gradient of displacement, in Zhang et al (1998) rotation gradients are considered, then adopting a couple stress theory which is the specialisation of the model adopted here when plasticity is absent.…”

The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results

“…Therefore, many efforts have been directed towards more or less mathematically rigorous homogenization procedures leading to this class of continua (e.g. [69][70][71][72][73][74][75]). In particular, it has been noted that the introduction of Nthorder models is suitable for describing non-local effects [76][77][78][79][80], some bio-mechanical phenomena [81][82][83][84][85][86][87][88], damage phenomena occurring in crack formation and growth (see those described in, for example, [89][90][91][92][93][94][95][96]) and internal friction in solids [97].…”