Theory of Couple-Stresses in Bodies with Constrained Rotations

Abstract

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On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush

“…• the indeterminate couple stress model (Grioli-Koiter-Mindlin-Toupin model) [9,2,19,28,42,40,10] in which the higher derivatives (apparently) appear only through derivatives of the infinitesimal continuum rotation curl u. Hence, the curvature energy has the equivalent forms…”

“…The indeterminate couple stress model is not simply obtained as a constraint Cosserat model [40], i.e. assuming that A = axl(skew∇u).…”

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

“…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.…”

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

“…These quantities are not independent, they are related by the equations of linear couple stress theory of elasticity [18,38,40]. Taking into account that the couple stress theory of curved rods will be studied here, the curvilinear orthogonal system of coordinates will be used for that.…”

“…Since the publication of the landmark book of Cosserat brothers [4] in literature there are various known generalizations and applications of the classical or Cauchy continuum which are summarized in many books [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and papers [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Especially we must mention remarkable books of Eringen [38,39] and Nowaski [40] and fundamental treatise of Truesdell and Toupin [3].…”