An Assessment of Couple Stress Theories

Abstract

In this paper, we examine the mathematical and physical consistencies of the three primary couple stress theories: original Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), modified couple stress theory (M-CST) and consistent couple stress theory (C-CST).  As has been known for many years, MTK-CST suffers from some fundamental inconsistencies, such as the indeterminacy of the couple-stress tensor.  Therefore, despite the fact that MTK-CST has a fundamental position in the evolution of size-dependent continuum mechanics, it is not a reliable theory within continuum mechanics, for example, in developing new size-dependent multi-physics formulations.  We also observe that M-CST not only inherits all inconsistences from the original MTK-CST, but also suffers from new additional inconsistencies, such as the introduction of a new non-physical governing equation.  These inconsistencies refute the claim of those who state that the couple-stress tensor may be chosen symmetric.  Therefore, the apparent success of MTK-CST and M-CST in describing a size-effect for some problems, such as two-dimensional plate and beam bending, is not enough to justify these theories as suitable for general cases.  In fact, the symmetric couple-stresses in M-CST create torsional or anticlastic deformation, not bending.  On the other hand, C-CST, with a skew-symmetric couple-stress tensor, is the consistent continuum mechanics suitable for solving different size-dependent solid, fluid and multi-physics problems. 

A novel upgraded molecular mechanics framework with considering the length scale parameter and its advantages in analyzing the behavior of nanostructures

“…Other computations must be made to estimate torsional stiffness. Considering the overall effect of modified couple stresses on torsional stiffness, the resultant torque at the ends of the bar seems to be 26 :…”

Bending characteristics of carbon nanotubes: Micropolar elasticity models and molecular dynamics simulations

“…As a result, micropolar theory has been successfully used to describe a wide range of materials with internal structures, especially those where relative rotations are predominant, such as brick masonry-like materials [59][60][61], heterogeneous structures with internal cracks/voids/inclusions [62][63][64][65], and particulate composites [66][67][68][69][70]. A subcase of micropolar theory is the couple stress theory originally developed by Toupin [71], Mindlin and Tiersten [72] and Koiter [73], in which microrotations are constrained to follow macrorotations (local rigid rotations) yielding symmetric strain measures [74,75], however, it should be noted that the original form of couple stress theory suffers from indeterminacy of the spherical part of couple stress tensor and the appearance of the normal component of couple traction vector on boundary surfaces [47,76,77]. To overcome this inconsistency, different versions of couple stress theory have been developed.…”

“…Yang et al proposed modified couple stress theory [78] in which the microrotations still equals the macrorotation while the strain energy is assumed to depend only on the symmetric part of the strain and curvature tensors, providing a symmetric couple stress tensor and only one additional length scale parameter besides classical material constants. Due to the greater simplicity of size-dependent modified couple stress theory than micropolar theory and original couple stress theory, it is frequently adopted to study the mechanical behavior of size-dependent materials and nanostructures [79][80][81][82][83], although there still exists some criticism on this theory regarding ill-posed boundary conditions [77].…”

A modified couple stress model to analyze the effect of size-dependent on thermal interactions in rotating nanobeams whose properties change with temperature

“…On the contrary, the Gurtin-Murdoch Surface Elasticity Theory utilizes three surface elasticity constants to simulate the influence of surface energy. Hadjesfandiari and Dargush's [16,17] investigation focused on examining the recently developed skew-symmetric CST to show its internal coherence, inherent simplicity, and fundamental relationship with the theories of classical physics.…”