A consistent variational theory of the higher–order nonlocal gradient elasticity is conceived to appropriately introduce the nonlocality to the higher-order strain gradient theory. The abstract variational approach, based on appropriate functional spaces of test fields, is applied to establish the higher–order nonlocal gradient mechanics of elastic beams in flexure. Two nonlocal and two gradient characteristic lengths are exploited to describe the size–dependent response of continua with nano–structures. Integral convolutions of the higher–order constitutive law are restored to the equivalent differential problem endowed with non–standard boundary conditions of constitutive–type. The higher–order strain gradient theory, higher–order nonlocal elasticity and modified nonlocal strain gradient theory, extensively adopted in the community of Engineering Science, are demonstrated to be particular cases of the introduced higher-order nonlocal gradient theory. The well-posedness of the developed higher-order nonlocal gradient problem is revealed by studying the flexural response of structures with wide-ranging applications in nano-engineering. Exact analytical solution for elastostatic deflections of nano-beams is derived and new benchmark examples of nano-mechanics interest are detected. The higher-order nonlocal gradient elasticity theory can effectively characterize nanoscopic phenomena in advanced nano–composites and nano–structures.

高阶非局部梯度弹性的一致变分理论被认为可以适当地将非局部性引入高阶应变梯度理论。基于测试场的适当功能空间的抽象变分方法，被用于建立挠曲弹性梁的高阶非局部梯度力学。利用两个非局部长度和两个梯度特征长度来描述纳米结构连续体的尺寸依赖性响应。高阶本构律的积分卷积被还原为本构型的非标准边界条件所赋予的等效微分问题。高阶应变梯度理论，高阶非局部弹性和修正的非局部应变梯度理论，在工程科学界广泛采用的方法被证明是引入的高阶非局部梯度理论的特殊情况。通过研究在纳米工程中具有广泛应用的结构的挠曲响应，揭示了已发展的高阶非局部梯度问题的适定性。推导了纳米束弹性挠度的精确解析解，并检测了纳米力学感兴趣的新基准示例。高阶非局部梯度弹性理论可以有效地表征高级纳米复合材料和纳米结构中的纳米现象。通过研究在纳米工程中具有广泛应用的结构的挠曲响应，揭示了已发展的高阶非局部梯度问题的适定性。推导了纳米束弹性挠度的精确解析解，并检测了纳米力学感兴趣的新基准示例。高阶非局部梯度弹性理论可以有效地表征高级纳米复合材料和纳米结构中的纳米现象。通过研究在纳米工程中具有广泛应用的结构的挠曲响应，揭示了已发展的高阶非局部梯度问题的适定性。推导了纳米束弹性挠度的精确解析解，并检测了纳米力学感兴趣的新基准示例。高阶非局部梯度弹性理论可以有效地表征高级纳米复合材料和纳米结构中的纳米现象。