The higher-order unified gradient elasticity theory is conceived in a mixed variational framework based on suitable functional space of kinetic test fields. The intrinsic form of the differential and boundary conditions of equilibrium along with the constitutive laws is consistently established. Various forms of the gradient elasticity theory, in the sense of stress or strain gradient models, can be retrieved as particular cases of the introduced generalized elasticity theory. The proposed stationary variational principle can effectively realize the nanoscopic structural effects while being exempt of restrictions typical of the nonlocal gradient elasticity model. The well-posed generalized gradient elasticity theory is invoked to study the mechanics of torsion and the torsional behavior of elastic nano-bars is analytically examined. The closed-form analytical formulae of the size-dependent shear modulus of nano-sized bar is determined and efficiently applied to reconstruct the shear modulus of SWCNTs with dissimilar chirality in comparison with the numerical simulation data. A practical approach to calibrate the characteristic lengths associated with the higher-order unified gradient elasticity theory is introduced. Numerical results associated with the torsion of higher-order unified gradient elastic bars are demonstrated and compared with the counterpart size-dependent elasticity theories. The conceived generalized gradient elasticity theory can beneficially characterize the nanoscopic response of advanced nano-materials.

高阶统一梯度弹性理论是在基于合适的动力学测试场函数空间的混合变分框架中构想的。平衡的微分和边界条件以及本构定律的内在形式是一致建立的。梯度弹性理论的各种形式，在应力或应变梯度模型的意义上，可以作为引入的广义弹性理论的特例进行检索。所提出的平稳变分原理可以有效地实现纳米级结构效应，同时不受非局部梯度弹性模型的典型限制。引用适定广义梯度弹性理论来研究扭转力学，并对弹性纳米棒的扭转行为进行分析检验。确定了纳米棒尺寸相关剪切模量的封闭式解析公式，并与数值模拟数据进行比较，有效地重建了具有不同手性的 SWCNT 的剪切模量。介绍了一种校准与高阶统一梯度弹性理论相关的特征长度的实用方法。证明了与高阶统一梯度弹性杆的扭转相关的数值结果，并与对应的尺寸相关弹性理论进行了比较。设想的广义梯度弹性理论可以有益地表征先进纳米材料的纳米级响应。