In classical mechanics, considering Hook’s law stress is a linear function of strain. While in strain gradient theory stress is a function of strain and strain differentials. In this paper, Novel formulation relating stress and strain and also new boundary conditions are derived based on minimum potential energy principle. In strain gradient theory a length coefficient parameter is defined. This statistical parameter shows that material behaviour in microscopic scale depends on material dimensions. In classical elasticity dependency of the material behaviour on material size could not be described due to the lack of length coefficient parameter. Here also a total stress tensor, different from the Cauchy’s stress tensor, is defined which can be used as a total stress tensor in momentum equation. Using strain gradient theory, strain field for a rotational shaft with a constant angular speed is analytically studied. Knowing displacement field, total stress tensor also can be computed. In the derived displacement field in addition to two Lame constants there is also a material constant. Formulations based on strain gradient theory turn to those of classical mechanics if length coefficient is neglected. Results of stress analysis using strain gradient theory and those of classic mechanics are compared.
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