Boundary modes localized at the edge or on the interface of topological mechanical lattices are analogous to their electronic counterparts in topological insulators and are robust and immune to structural imperfections. Most studies on the boundary modes in mechanical lattices are based on band theories, focusing on amplitude-independent behaviors at different frequency ranges. However, highly distorted topological mechanical lattices exhibit amplitude-dependent nonlinear effects that are difficult to characterize. In this study, a topological mechanical lattice exhibiting amplitude-dependent boundary modes is explored to uncover the evolution and connection between boundary modes in the linear and nonlinear regimes. Both discrete and continuum models are developed. An iterative method for the discrete model is proposed to numerically capture the boundary mode evolution while, based upon the continuum model, an analytical solution of the decay length scale of the boundary modes is obtained and can reduce to the linear band theory prediction at the small amplitude limit. A design strategy is presented for programmable topological polarization in the lattice that can be used to encode mechanical information. The obtained results provide insight into the modulation of topological protected mechanical response and stimulation of spatially ordered modes.

定位在拓扑机械格的边缘或界面处的边界模式类似于它们在拓扑绝缘体中的电子对应物，并且健壮并且不受结构缺陷的影响。对机械晶格中边界模式的大多数研究都是基于能带理论的，重点是在不同频率范围内与振幅无关的行为。但是，高度变形的拓扑机械晶格表现出幅度依赖的非线性效应，难以表征。在这项研究中，探索了一种表现出幅度依赖的边界模式的拓扑机械格，以揭示线性和非线性状态下边界模式之间的演化和联系。开发了离散模型和连续模型。提出了一种离散模型的迭代方法，以数值方式捕获边界模态的演化，同时，在连续模型的基础上，获得了边界模态的衰变长度尺度的解析解，可以将其简化为线性带理论的预测。幅度极限。提出了一种设计策略，用于晶格中的可编程拓扑极化，可用于对机械信息进行编码。获得的结果提供了对拓扑受保护的机械响应的调制和对空间有序模式的刺激的见解。提出了一种设计策略，用于晶格中的可编程拓扑极化，可用于对机械信息进行编码。获得的结果提供了对拓扑受保护的机械响应的调制和对空间有序模式的刺激的见解。提出了一种设计策略，用于晶格中的可编程拓扑极化，可用于对机械信息进行编码。获得的结果提供了对拓扑受保护的机械响应的调制和对空间有序模式的刺激的见解。