Elasticity solutions of inhomogeneous and anisotropic nano-circular rings

This study extends classical elasticity to gradient elasticity by investigating the analytical solutions for inhomogeneous and anisotropic curvilinear nano-beams with axial symmetry. For this purpose, we consider two variations for the elastic material coefficients along the thickness of the curvilinear beam. First, the coefficients are assumed to be proportional to the radial coordinate as $s_{ij}\left(r\right)=s_{ij}r$. Secondly, it is assumed that the coefficients are linear functions of the radial coordinate with two coefficients of the material coefficients such as $s_{ij}\left(r\right)=s_{ijc}+s_{ijg}r$. For both cases of variation of the elastic coefficients, the analytical solutions of stress fields for both classical and nano-curvilinear beams are obtained by using the definition of the gradient Airy stress function introduced for the gradient elasticity theory, similar to the Airy stress function notation defined in the classical elasticity theory. Then, analytical solutions of displacement fields are given similarly for classical and nano-curvilinear beams. As a special application of this general case, circular rings’ stress and displacement fields subjected to internal and external pressures are examined for the classical and nano-beam cases. Furthermore, the initial stress fields, depending on the initial pressure, are examined in the classical and gradient elasticity theory using the notation of the initial gradient pressure and initial gradient stress fields. Lastly, an expansion for the small gradient coefficient $c\ll 1$ is performed analytically, as the solutions presented are otherwise numerically difficult to evaluate within this regime. The expansion allows us to show analytically that for all derived stress and displacement fields, including the gradient Airy stress functions, the gradient elasticity solutions converge to the classical elasticity as the gradient coefficient $c$ goes to zero.