AI 中文总结
研究非欧几里得带中的形状转变问题，借助\(\Gamma\)-收敛从三维模型推导出一维极限理论，结合多种技术，证明了带中形状转变的存在，推广相关结果以更准确模拟物理系统。
AI 中文摘要
带是厚度\(t\)远小于宽度\(w\)，宽度又远小于长度的薄弹性体。我们从三维模型出发，借助\(\Gamma\)-收敛推导出具有粗糙预应变的带的一维极限理论。模型表明，窄带的能量极小值与中线处的参考（有效）第二基本形式一致，宽带则不然。这证明了此类带中形状转变的存在，推广了Maor和Mora的结果，使其适用于粗糙预应变，能更准确地模拟许多相关物理系统。我们的分析结合了Freddi等人对欧几里得带的研究技术、Schmidt对预应变板降维的工作以及Maor和Mora关于缩放极限应变的新结构定理。
英文摘要
Ribbons are thin elastic bodies whose thickness $t$ is much smaller than their width $w$, which is in turn much smaller than their length. Starting from a three-dimensional model, we derive a one-dimensional limit theory for ribbons with rough prestrain, by means of $Γ$-convergence. Our model shows that for narrow ribbons, the energy minimizer coincides with the reference (effective) second fundamental form along the midline, while for wide ribbons this is generically not the case. This proves the existence of shape-transitions in such ribbons, as observed in many experiments, generalizing recent results by Maor & Mora to rough prestrains, which more accurately model many of the relevant physical systems. Our analysis combines techniques from the study of Euclidean ribbons due to Freddi et al., the work of Schmidt on dimension reduction of prestrained plates, and a new structure theorem on the scaled limiting strain due to Maor & Mora.
CommentsThis work was part of a master's thesis at The Hebrew University of Jerusalem, done under supervision of Prof. Cy Maor
