完整二维二阶梯度连续介质的合成：三 pantographic 织物
On the synthesis of complete two-dimensional second-gradient continua: Tri-pantographic fabrics
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中文总结 AI 辅助
本研究提出二维二阶梯度弹性连续介质的完备性概念，通过微结构路径合成，证明三 pantographic 连续介质完备，揭示微结构结构对连续介质完备性及高阶边界相互作用的影响。
中文摘要 AI 辅助
我们提出二维二阶梯度弹性连续介质的完备性概念，并给出其合成的微结构路径。若连续介质关于二阶梯度变量的储能的 Hessian 局部正定，则称其为完备的，即对每个构型，位置二阶梯度的每个非零容许增量在能量的最高阶部分中受二次控制。从虚功原理出发，我们推导了一类广泛的纤维二阶梯度连续介质的本构关系、平衡方程和容许边界相互作用，这类介质的储能依赖于纤维拉伸、拉伸梯度和曲率。该理论随后被应用于多个由 pantographic 微结构驱动的连续介质：经典 pantographic 薄板被证明是不完备的，双 pantographic 织物扩大了能量可检测的二阶梯度分量类别但仍不完备；最后，我们构建了与所提出的三族结构相关的三 pantographic 连续介质，并证明其是完备的。这些例子说明微结构结构如何影响等效连续介质的完备性性质及其高阶边界相互作用的逐点形式。
英文摘要
We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.
