Ciarletta P, Pozzi G, Riccobelli D
MOX - Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy.
R Soc Open Sci. 2022 May 18;9(5):220421. doi: 10.1098/rsos.220421. eCollection 2022 May.
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 us 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 four-dimensional printing.
最初，应力板因其通过外部刺激可调节形态，而被广泛应用于现代制造技术中，如增材制造和紫外光刻。在这项工作中，我们利用具有初始应力的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，对具有初始应力的弹性板的Föppl-von Kármán方程进行了形式渐近推导。与现有工作相比，我们的方法使我们能够在不规定弹性体无应力状态的潜在目标度量的情况下，确定弹性板的形态转变。我们在一些具有工程意义的物理问题中明确求解了导出的FvK方程，讨论了初始应力分布如何驱动变形板内自发曲率的出现。所提出的数学框架可用于按需定制形状，应用于从软体机器人到四维打印等多个工程领域。
