Stationary variational principle of mixture unified gradient elasticity

The stress gradient, strain gradient, and classical elasticity theory are integrated within a consistent variational framework to conceive the mixture unified gradient theory of elasticity. The significant advantage of the established stationary variational principle lies in incorporating all the governing equations, viz. the boundary-value problem of the associated dynamic equilibrium along with the non-classical boundary conditions and the constitutive laws, into a single functional. The mixture unified gradient theory can effectively serve as a suitable counterpart for the two-phase local/nonlocal gradient theory with a noteworthy privilege; the conceived augmented elasticity theory can be efficiently adopted to examine various multi-dimensional structural problems of practical interest in Engineering Science. The well-posedness of the introduced generalized size-dependent elasticity theory is demonstrated via analytically examining the flexure mechanics of inflected elastic nano-beams. A viable approach is proposed for calibrating the gradient length-scale parameters associated with the mixture unified gradient elasticity, and accordingly, the size-dependent Young's modulus of carbon nanotubes can be inversely determined. The established size-dependent elasticity theory can be fruitfully invoked to address problems in nano-mechanics practice where the mechanical response is notably affected by nano-structural features. The developed mixture unified gradient theory of elasticity can, hence, pave way ahead in mechanics of ultra-small structures.