A plethora of current applicative problems of Nano-Engineering involve nanostructures interacting with elastic media which exhibit significant size effects. Internal elasticity of the nanostructure and external elasticity of the surrounding foundation can be conveniently modelled by two nonlocal continua, provided that the relevant structure-soil problem results to be well-posed. This issue is still an open problem of Engineering Science and is solved in the present study by improving the Eringen–Wieghardt integral elasticity theory for nanobeams on nano-foundations, without having to resort to local/nonlocal mixtures, governed by a higher number of scale parameters, or physically questionable artifices to get mathematical consistency. Nanobeams resting on nano-foundations are modelled by stress-driven and displacement-driven laws providing nonlocal curvatures and reactions by convolution integrals of bending moments and displacements with bi-exponential averaging kernels. Advantageously, the relevant integro-differential problem is reverted to a simpler differential formulation equipped with non-classical boundary conditions. Effectiveness of the proposed nonlocal approach is illustrated by analytically solving beam–soil systems simulating nanocomposites and biological cells.

大量当前的纳米工程应用问题涉及与弹性介质相互作用的纳米结构，这些弹性介质表现出显着的尺寸效应。纳米结构的内部弹性和周围地基的外部弹性可以方便地通过两个非局部连续体来模拟，前提是相关的结构-土壤问题是适定的。这个问题仍然是工程科学的一个悬而未决的问题，在本研究中通过改进纳米基础上纳米梁的 Eringen-Wieghardt 积分弹性理论来解决，而不必诉诸局部/非局部混合物，由更高数量的尺度控制参数或物理上有问题的技巧以获得数学一致性。基于纳米基础的纳米梁通过应力驱动和位移驱动定律建模，通过弯矩和位移的卷积积分与双指数平均内核提供非局部曲率和反应。有利地，相关的积分-微分问题被恢复为配备有非经典边界条件的更简单的微分公式。通过分析求解模拟纳米复合材料和生物细胞的梁土系统来说明所提出的非局部方法的有效性。