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.

将应力梯度、应变梯度和经典弹性理论整合在一个一致的变分框架内，构想出混合统一的弹性梯度理论。已建立的平稳变分原理的显着优势在于结合了所有控制方程，即。将相关动态平衡的边值问题与非经典边界条件和本构定律结合为一个单一的泛函。混合统一梯度理论可以有效地作为两相局部/非局部梯度理论的合适对应物，具有值得注意的特权；构想的增强弹性理论可以有效地用于研究工程科学中具有实际意义的各种多维结构问题。通过分析检查弯曲弹性纳米梁的弯曲力学，证明了引入的广义尺寸相关弹性理论的适定性。提出了一种可行的方法来校准与混合物统一梯度弹性相关的梯度长度尺度参数，因此可以反向确定碳纳米管的尺寸相关杨氏模量。已建立的尺寸相关弹性理论可以有效地用于解决纳米力学实践中的问题，其中机械响应受到纳米结构特征的显着影响。因此，发展起来的混合统一弹性梯度理论可以为超小型结构的力学铺平道路。