从宇宙学到凝聚态，局域态在自然界中普遍存在，是自然和社会科学众多分支中的热点。这其中，对固体/结构中局域变形的理解和调控是力学乃至工程科学中的经典话题。然而，长期以来人们对此形成的一个普遍认知是：局域变形对缺陷敏感，使得其位置和尺度随机而难以预测。力学超材料作为可设计的人工介质，通常表现出非同寻常的物理力学性质，为应对这一挑战提供了全新机遇。本文从结构相变的视角，研究了力学超材料拓扑结构的变形演化，构建了可实现编程有序局域变形的静态孤子普适性框架，揭示了边界可控局域变形的物理机制。超材料拓扑结构与力学响应调控。本文采用实空间中的几何参数，对所关注的超材料基元进行了拓扑分类。通过调整该基元的序参量，实现了超材料元胞“单—双—单”稳态的调控和可编程的“负—零—正”泊松比，进而实现新颖的“均匀—规则局域化—均匀”的变形演化。理论模型揭示了分类与调控的本质：该几何参数决定了元胞原位势的分岔以及系统控制方程对称性，且均匀变形对应于控制方程平凡解。三类不同的变形模式，分别倾向于关闭、打开和拓宽超材料初始构型的低频禁带。有序局域变形与静态周期孤子的激发。本文通过试验研究，激发出具有静态周期孤子特性的可控有序局域化变形，进而提出了编程设计此类变形的普适性框架。通过引入结构相变与元胞极化的概念，建立理论模型，证实了这些未曾意料的孤子，源自于超材料型相变中的周期性拓扑激发，可由著名的$\phi^4$理论所描述。该理论框架的普适性植根于，这一激发并不依赖于特定的元胞类型，而取决于元胞原位势与强耦合能的竞争。此外，所发现的孤子数与波长的尺寸效应揭示了有限尺寸力学超材料中的两个内蕴特征尺度。边界可控的局域变形与结构相变。本文通过双轴试验和数值模拟研究，发现边界约束可进一步调控局域变形的激发。在比例位移载荷下，通过改变应变比可实现静态畴壁的萌生、翻转与湮灭，进而实现对局域变形的调控。应变约束的符号决定了畴壁特征，且零应变对应畴壁翻转的临界点。为揭示其潜在的物理机制，基于Landau理论，建立了非对称$\lambda\phi^4$模型，诠释了静态畴结构的演化过程，揭示了边界约束的角色：调控原位势的对称性，进而决定超材料相变类型与畴壁激发。
Localized states are ubiquitous in physics, ranging from cosmology to condensed matter. Among them, understanding and regularizing deformation localization in solids/structures are a classic topic of mechanics and even engineering science. However, for a long time, a common perception about this is that localized deformations are sensitive to inherent or externally induced defects. Consequently, it is a widely recognized challenge to predict the precise location and size of localization zones. As an engineered matter, mechanical metamaterials usually exhibit exotic physical and mechanical properties, which provides brand-new opportunities to target the challenge. This dissertation, from the perspective of structural phase transition, aims at systematically engineering the pattern-morphing (i.e., topology) of mechanical metamaterial, constructing a general static-soliton framework for programming ordered deformation localization, and unveiling the physical mechanism of boundary-controllable localized deformations. Metamaterial topology and mechanical response regulation. We use a geometric parameter in real space to topologically classify the unit cell of the considered metamaterial. By altering the parameter, the states of the unit cell, from monostable to bi-stable and then back to monostable, can be programmed. Under uniaxial quasi-static compression, we can program not only ‘negative-zero-positive’ Poisson ratios, but also the novel ‘homogenous-ordered localized-homogenous’ deformed configurations. Our theoretical model reveals the nature of the classification and regulation: the geometric parameter dictates the bifurcation of the onsite potential landscape, as well as the symmetry of the system’s equilibrium equation. Therein, the macro-homogenous patterns correspond to trivial solutions of the governing equation. We observe that distinct pattern-transition processes influence the elastic waves remarkably, including closing, opening, and broadening low-frequency band gaps.Ordered deformation localization and static periodic-soliton. We experimentally excite an ordered deformation localization in the mechanical metamaterial in a controllable manner, and construct a general static-soliton framework for programming such patterns. We introduce structural phase transition to the mechanical metamaterial, and harness the outward/inward polarizations to the bistable unit cells. Our theoretical model shows that these unanticipated solitons stem from displacive phase transitions with periodic topological excitations captured by the well-known $\phi^4$ theory. The programmability and generality of the physical framework root in that the periodic excitations do not arise from the specific architecture of the repeating structural motifs, but rather from the interactions between the onsite potentials and strong coupling energies of unit cells. Moreover, the size-dependence of soliton number and wavelength demonstrates two intrinsic length scales in the finite-size kink lattices, which can be employed to quantify the Saint-Venant Principle in mechanics.Boundary-controllable localization and structural phase transition. Our experiments and simulations confirm that boundary constraints can further regulate deformation localization. Under displacement-controlled proportional loadings via tuning the strain ratio $\varepsilon_x/\varepsilon_y$, we can program the creation, reversal and annihilation of static domain walls in metamaterials, and ensure feasible control of localizations. It is found that the sign of $\varepsilon_x$ determines the domain wall type, and $\varepsilon_x = 0$ is the critical point of domain flipping. To reveal the underlying physical mechanism, an asymmetric $\lambda\phi^4$ model is established based on the Landau theory. The model faithfully captures the evolution of the domain structure and uncovers the role of the boundary constraint $\varepsilon_x$: controlli\label{key}ng the symmetry of the onsite potential which in turn determines the phase transition type and domain wall excitations in metamaterials.
