We investigate energetically optimal configurations of thin structures with a pre-strain. Depending on the strength of the pre-strain we consider a whole hierarchy of effective plate theories with a spontaneous curvature term, ranging from linearised Kirchhoff to von Kármán to linearised von Kármán theories. While explicit formulae are available in the linearised regimes, the von Kármán theory turns out to be critical and a phase transition from cylindrical (as in linearised Kirchhoff) to spherical or saddle-shaped (as in linearised von Kármán) configurations is observed there. We analyse this behaviour with the help of a whole family ( I vK θ ) θ ∈ ( 0 , ∞ ) $(\mathcal{I}^{\theta }_{\mathrm{vK}})_{\theta \in (0,\infty )}$ of effective von Kármán functionals which interpolates between the two linearised regimes. We rigorously show convergence to the respective explicit minimisers in the asymptotic regimes θ → 0 $\theta \to 0$ and θ → ∞ $\theta \to \infty $ . Numerical experiments are performed for general θ ∈ ( 0 , ∞ ) $\theta \in (0,\infty )$ which indicate a stark transition in a critical region of parameters θ $\theta $ .

中文翻译：

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预应变多层的能量最小化配置

我们研究了具有预应变的薄结构的能量优化配置。根据预应变的强度，我们考虑具有自发曲率项的有效板块理论的整个层次结构，范围从线性化基尔霍夫理论到冯卡门理论再到线性化冯卡门理论。虽然在线性化状态下可以使用显式公式，但事实证明 von Kármán 理论是关键的，并且在那里观察到从圆柱形（如线性化基尔霍夫）到球形或鞍形（如线性化 von Kármán ）配置的相变。我们在整个家庭的帮助下分析这种行为 ( I vK θ ) θ ∈ ( 0 , ∞ ) $(\mathcal{I}^{\theta }_{\mathrm{vK}})_{\theta \in (0,\infty )}$ 有效的 von Kármán 泛函，它在两个线性化机制之间进行插值。我们严格地证明在渐近状态 θ → 0 $\theta \to 0$ 和 θ → ∞ $\theta \to \infty $ 中收敛到各自的显式极小值。对一般 θ ∈ ( 0 , ∞ ) $\theta \in (0,\infty )$ 进行了数值实验，这表明参数 θ $\theta $ 的临界区域中存在明显的转变。