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In this paper we venture a new look at the linear isotropic indeterminatecouple stress model in the general framework of second gradient elasticity andwe propose a new alternative formulation which obeys Cauchy-Boltzmann's axiomof the symmetry of the force stress tensor. For this model we prove theexistence of solutions for the equilibrium problem. Relations with othergradient elastic theories and the possibility to switch from a {4th order}(gradient elastic) problem to a 2nd order micromorphic model are also discussedwith a view of obtaining symmetric force-stress tensors. It is shown that theindeterminate couple stress model can be written entirely with symmetricforce-stress and symmetric couple-stress. The difference of the alternativemodels rests in specifying traction boundary conditions of either rotationaltype or strain type. If rotational type boundary conditions are used in thepartial integration, the classical anti-symmetric nonlocal force stress tensorformulation is obtained. Otherwise, the difference in both formulations is onlya divergence--free second order stress field such that the field equations arethe same, but the traction boundary conditions are different. For these resultswe employ a novel integrability condition, connecting the infinitesimalcontinuum rotation and the infinitesimal continuum strain. Moreover, we providethe complete, consistent traction boundary conditions for both models.

A variant of the linear isotropic indeterminate couple stress model with  symmetric local force-stress, symmetric nonlocal force-stress, symmetric  couple-stresses and complete traction boundary conditions