Exact solutions for elastic response in micro- and nano-beams considering strain gradient elasticity

Abstract

This paper presents the exact solutions derived for the static bending response of simply supported isotropic micro- and nano-beams. The governing differential equations of equilibrium and the associated boundary conditions for the beam are derived by applying the variational principle over the internal energy functional involving strain and strain gradient terms. The Mindlin’s Form II model for higher order metrics of energy involving strain gradient terms and three additional material length constants has been used for the current formulation. Constitutive relations for stress and the higher order stress in an isotropic solid, which are the conjugates of strain and strain gradient, respectively, are provided in this paper. The exact solutions for the displacement vector are obtained by solving the governing differential equations. The significance of the gradients and the corresponding conjugates are studied using numerical examples of micro- and nano-beams. These results are compared with those obtained from the classical model of elasticity for structures which does not involve strain gradients. Due to the consideration of strain gradients, the structures exhibit a significant increase in stiffness as the beam dimensions are reduced. The profiles for stress developed across the thickness are presented to study the change in the behavior of the structures at a micro- and nano-level. The importance of considering strain gradients for the design of low-dimensional structures in sensitive applications is noted. The exact solutions developed in this paper can be used as benchmark solutions for validating further research on the behavior of low-dimensional bodies using computational models and experiments. The present study calls for the development of new failure theory of isotropic materials possessing strain gradients.

Keywords

Strain gradient elasticity micro and nano beams exact solutions size-effect energy methods

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