非线性夹杂理论及其在约束体生长和形态发生中的应用
Nonlinear Inclusion Theory with Application to the Growth and Morphogenesis of a Confined Body
摘要：
对材料力学行为研究最著名的贡献之一是J. D. Eshelby在50年代后期彻底改变了我们对弹性应力和应变场的理解，这是由于椭圆形夹杂物/不均匀性经历了形状和尺寸的转变。虽然Eshelby的工作为各个领域的重大进步奠定了基础，包括断裂力学、相变理论和均匀化方法，但其扩展到大变形范围以及材料可以主动重组的情况。响应有限的转变应变，处于新生状态。除了高度非线性材料响应所带来的理论困难之外，一个主要的障碍是缺乏实验观察，可以阐明这种机制中出现的复杂性。为了解决这一限制，我们的实验观察揭示了嵌入水凝胶中的霍乱弧菌生物膜的关键形态发生步骤，因为它们从初始大小增长了四个数量级。使用生物膜生长作为案例研究，我们的理论模型考虑了各种生长情况，并采用两种不同的和互补的方法-最小的分析模型和有限元计算-获得近似平衡的解决方案。一个特别强调的是把确定的自然生长路径的夹杂物，优化其形状响应的限制，并在矩阵中，这一起解释所观察到的行为的生物膜的损伤的发病。除了细菌生物膜，这项工作揭示了力学在确定受限生长体的形态发生途径中的作用，因此适用于自然和工程材料系统中普遍存在的广泛现象。
英文摘要：
One of the most celebrated contributions to the study of the mechanical behavior of materials is due to J.D. Eshelby, who in the late 50s revolutionized our understanding of the elastic stress and strain fields due to an ellipsoidal inclusion/inhomogeneity that undergoes a transformation of shape and size. While Eshelby's work laid the foundation for significant advancements in various fields, including fracture mechanics, theory of phase transitions, and homogenization methods, its extension into the range of large deformations, and to situations in which the material can actively reorganize in response to the finite transformation strain, is in a nascent state. Beyond the theoretical difficulties imposed by highly nonlinear material response, a major hindrance has been the absence of experimental observations that can elucidate the intricacies that arise in this regime. To address this limitation, our experimental observations reveal the key morphogenesis steps of Vibrio cholerae biofilms embedded in hydrogels, as they grow by four orders of magnitude from their initial size. Using the biofilm growth as a case study, our theoretical model considers various growth scenarios and employs two different and complimentary methods -- a minimal analytical model and finite element computations -- to obtain approximate equilibrium solutions. A particular emphasis is put on determining the natural growth path of an inclusion that optimizes its shape in response to the confinement, and the onset of damage in the matrix, which together explain the observed behavior of biofilms. Beyond bacterial biofilms, this work sheds light on the role of mechanics in determining the morphogenesis pathways of confined growing bodies and thus applies to a broad range of phenomena that are ubiquitous in both natural and engineered material systems.
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