The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

弹性的混合应力梯度理论是通过在平稳变分框架内一致地统一经典弹性理论和应力梯度理论而构想的。与功能梯度纳米棒相关的边界值问题是严格制定的。确定了轴向力场的本构律，并配备了适当的非标准边界条件。通过对纳米力学中具有实际意义的结构方案的轴向位移场的分析分析，证明了在有限结构域上定义的混合应力梯度问题的适定性证据。此外，一种有效的无网格数值方法是 基于所提出的平稳变分原理引入，同时采用运动学和动力学场变量的自主级数解。坐标函数的合适数学形式根据修正的切比雪夫多项式提出，满足所需的经典和非标准边界条件。在混合参数和应力梯度特征参数方面，功能梯度纳米棒的轴向位移场的数值结果与解析解对应物的数值结果在纳米棒的整个跨度上得到了很好的一致性。因此，确保了已建立的无网格数值方法的有效性，证明了快速收敛速度和可接受的收敛区域。