具有初应力的弹性板的Föppl-von Kármán方程
The Föppl-von Kármán equations of elastic plates with initial stress
摘要：
初始应力板由于其通过施加外部刺激的可调形态而广泛用于现代制造技术中，例如增材制造和UV光刻。本文利用非线性弹性固体的本构理论，在不可压缩性和材料各向同性的假设下，给出了具有初应力的弹性板的Föppl-von Kármán方程的形式渐近推导。与现有的工作相比，我们的方法允许确定的弹性板的形态过渡，而不规定的基本目标度量的弹性体的无应力状态。我们显式地解决导出的FvK方程在一些物理问题的工程利益，讨论如何初始应力分布驱动变形板内的自发曲率的出现。所提出的数学框架可用于按需定制形状，应用于从软机器人到4D打印等多个工程领域。
英文摘要：
Initially stressed plates are widely used in modern fabrication techniques, such as additive manufacturing and UV lithography, for their tunable morphology by application of external stimuli. In this work, we propose a formal asymptotic derivation of the Föppl-von Kármán equations for an elastic plate with initial stresses, using the constitutive theory of nonlinear elastic solids with initial stresses under the assumptions of incompressibility and material isotropy. Compared to existing works, our approach allows to determine the morphological transitions of the elastic plate without prescribing the underlying target metric of the unstressed state of the elastic body. We explicitly solve the derived FvK equations in some physical problems of engineering interest, discussing how the initial stress distribution drives the emergence of spontaneous curvatures within the deformed plate. The proposed mathematical framework can be used to tailor shape on demand, with applications in several engineering fields ranging from soft robotics to 4D printing.
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