A new microstructure-dependent Saint–Venant torsion model based on a modified couple stress theory

Abstract

In this paper a new modified couple stress model is developed for the Saint-Venant torsion problem of micro-bars of arbitrary cross section. The proposed model is derived from a modified couple stress theory and has only one material length scale parameter.Using a variational procedure the governing differential equation and the associated boundary conditions are derived in terms of the warping function. This is a fourth order partial differential equation representing the analog of a Kirchhoff plate having the shape of the cross section and subjected to a uniform tensile membrane force with mixed Neumann boundary conditions. Since the fundamental solution of the equation is known, the problem could be solved using the direct Boundary Element Method (BEM). In this investigation, however, the Analog Equation Method (AEM) solution is applied and the results are cross checked using the Method of Fundamental Solutions (MFS). Several micro-bars of various cross-sections are analyzed to illustrate the applicability of the developed model and to reveal the differences between the current model and an existing one which, however, contains two additional constants related to the microstructure. Moreover, useful conclusions are drawn from the micron-scale torsional response of micro-bars, giving thus a better insight in the gradient elasticity approach of the deformable bodies.

Effect of size dependency on in-plane vibration of circular micro-rings

“…Outcome of these investigations a rms the size-dependent behavior of these components, and shows that torsional sti ness of micro-wires and bending sti ness of polymeric cantilevers are much more than the sti ness predicted by the classical theory. The size dependency in mechanical behavior of micro structures was also reported by other researchers [11][12][13].…”

Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua

“…The indeterminate couple stress model [19,22,32,48,55,68,69] appears by letting formally the Cosserat couple modulus µ c → ∞. This implies the constraint A = skew Du ∈ so(3).…”

Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

“…The indeterminate couple stress model [54][55][56][57][58][59][60] appears by letting formally the Cosserat couple modulus μ c → ∞. This implies the constraint A = skew Du ∈ so(3).…”

“…Shaat [56] used the micro-strain model with mixed terms and a degenerate curvature expression in DS, omitting S…”