Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity

Abstract

The stress-invariance problem for a chiral (non-centrosymmetric) micropolar material model is explored in two different planar problems: the in-plane and the anti-plane problems. This material model grasps direct coupling between the Cauchy-type and Cosserat-type (or micropolar) effects in Hooke's law. An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subjected to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently of the microstructure, geometry and phase distribution. These analytical results constitute a valuable means to validate computational procedures that handle this particular type of material model.

Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices

“…micropolar (Cosserat) theory (Cosserat E. and Cosserat F., 1909;Eringen, 1966). A general isotropic chiral (also known as non-centrosymmetric, acentric or hemitropic) micropolar theory introduces three additional material constants compared to the non-chiral theory, the additional material parameters change their signs according to the handedness of the microstructure to represent the effect of chirality (Nowacki, 1986;Lakes and Benedict, 1982;Lakes, 2001;Natroshvili and Stratis, 2006a,b;Joumaa and Ostoja-Starzewski, 2011).…”

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.…”

Green’s functions and Eshelby tensors for an ellipsoidal inclusion in a non-centrosymmetric and anisotropic micropolar medium

“…Another line of progress is related to exact connections or theories of micropolar elasticity. Among them, the results include the analogy between micropolar continuum and grid framework under initial stress (Bazant and Christensen, 1972), the CLM theorem (Cherkaev et al 1992) in planar Cosserat elasticity (Joumaa and Ostoja-Starzewski 2011, Jasiuk and Ostoja-Starzewski 2003, and Ostoja-Starzewski and Jasiuk 1995, the framework for effective medium of Cosserat medium (Yuan and Tomita 2001), conservation laws for micropolar elasticity (Lubarda and Markenscoff 2003), Hill's lemma in Cosserat media (Li and Liu 2009). Composites with proper microstructure could give rise to chiral effects in an overall sense.…”