The paper is focused on the dynamic homogenization of lattice-like materials with lumped mass at the nodes to obtain energetically consistent models providing accurate descriptions of the acoustic behavior of the discrete system. The equation of motion of the Lagrangian one-dimensional lattice is transformed according to a unitary approach aimed to identify equivalent non-local continuum models of integral-differential and gradient type, the latter obtained through standard or enhanced continualization. The bilateral Z-transform of the difference equation of motion of the lattice, mapped on the unit circle, is matched to the governing integral-differential equation of the equivalent continuum in the transformed Fourier space, which has the same frequency band structure as the Lagrangian one. Firstly, it is shown that the approximation of the kernels via Taylor polynomials leads to the differential field equations of higher order continua endowed with non-local constitutive terms. The field equations derived from such approach corresponds to the ones obtained through the so called standard continualization. However, the differential problem turns out to be ill-posed because the non-positive definiteness of the potential energy density of the higher order continuum. Energetically consistent equivalent continua have been identified through a proper mapping correlating the transformed macro-displacements in the Fourier space with a new auxiliary regularizing continuum macro-displacement field in the same space. Specifically, the mapping here introduced has zeros at the edge of the first Brillouin zone. The integral-differential governing equation and the corresponding differential one has been reformulated through an enhanced continualization, that is characterized by energetically consistent differential equations with inertial and constitutive non-localities. Here, the constitutive and inertial kernels of the integral-differential equation exhibit polar singularities at the edge of the first Brillouin zone. The proposed approach is generalized in a consistent way to two-dimensional lattices by using multidimensional Z- and Fourier transforms, a procedure that may be easily extended to three-dimensional lattices. Finally, two examples of lattice-like systems consisting of periodic pre-stressed cable-nets of point mass at the nodes are analyzed. The resulting gradient continuum models provide dispersion curves who are in excellent agreement with those of the Lagrangian systems.

本文的重点是节点上具有集中质量的格状材料的动态均质化，以获得能量上一致的模型，从而提供对离散系统声学行为的准确描述。拉格朗日式一维晶格的运动方程根据一种统一的方法进行了转换，旨在识别等效的积分微分和梯度类型的非局部连续模型，后者通过标准或增强的连续化获得。映射到单位圆上的晶格运动差分方程的双边Z变换与变换傅立叶空间中等效连续体的支配积分微分方程匹配，该傅立叶空间具有与拉格朗日函数相同的频带结构一。首先，结果表明，通过泰勒多项式逼近核，可以得到具有非局部本构项的高阶连续群的微分场方程。从这种方法得出的场方程对应于通过所谓的标准连续化获得的场方程。然而，由于高阶连续体的势能密度的非正定性，微分问题变得不适。通过将傅立叶空间中变换后的宏位移与同一空间中新的辅助正则化连续体宏位移场相关联的适当映射，可以确定出能量上一致的等效连续体。具体而言，此处介绍的映射在第一个布里渊区的边缘具有零。积分微分控制方程和相应的微分方程已通过增强的连续化进行了重新公式化，其特征在于具有惯性和本构非局部性的能量一致的微分方程。在这里，积分微分方程的本构和惯性核在第一个布里渊区的边缘显示出极性奇异性。通过使用多维Z-和傅立叶变换，可以以一致的方式将提出的方法推广到二维晶格，该过程可以轻松扩展到三维晶格。最后，分析了由节点处的点质量的周期性预应力电缆网组成的格状系统的两个示例。