This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

本文是处理n -Helmholtz型非局部弹性的广义理论的系列文章中的第二篇。该术语受以下事实的启发：整数型非局部本构方程的衰减函数（核）是与n级广义Helmholtz微分算子相关的Green函数。。在第一篇论文中，已经推导了控制方程，并由适当的热力学参数支持。在第二篇论文中，所提出的非局部模型专门用于解决一维边值问题的一维情况。首先，推导该理论的微分（或更确切地说是积分微分）版本中的相关高阶非标准边界条件。这些边界条件与积分公式中采用的特定衰减函数族一致。然后，介绍了一些简单的静力学和动力学问题的应用。特别是，该理论用于捕获静态响应并执行具有周期性微结构（质量-弹簧链）的离散晶格模型的自由振动分析，该周期性结构具有最近邻居和下一个最近邻居粒子之间的相互作用。在后一种情况下，边界效应出现在两个晶格末端，这些边界被建议的非局部连续谱公式很好地捕获了。通过匹配离散晶格模型的色散曲线，先验地识别出非局部材料参数，并就衰减函数进行了比较。