Determination of the Spherical Couple-Stress in Polar Linear Isotropic Elasticity

Abstract It is well known that the conventional couple-stress theory leaves the spherical part of the couple-stress indeterminate. This indeterminacy problem is recently resolved for fibrous composites subjected to either small or large deformations and containing a single family of fibres resistant in bending (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595). However, the problem remains still unsolved in simpler cases where the implied preference material direction is not related to fibre bending resistance, and even in the simplest possible case where the polar material of interest is linearly elastic and isotropic. This communication aims (i) to show that a relevant virtual spin concept employed (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595) is further applicable in the latter case of polar linear isotropic elasticity, (ii) to demonstrate the process in which that concept thus leads to determination of the spherical part of the couple-stress, (iii) to exemplify this process by providing a couple of simple illustrative examples, (iv) to specify and discuss the reason that the outlined method meets a hurdle in cases of linear anisotropic elasticity that is due to one or more preferential material directions, and, hence, (v) to further discuss the manner in which that newly identified difficulty is currently confronted, and may thus be handled successfully.