[Paper Review] Periodically wrinkled plate of the Föppl-von Kármán type

Igor Velčić|arXiv (Cornell University)|Apr 4, 2011

Structural Analysis and Optimization11 references3 citations

TL;DR

This paper derives a periodically wrinkled plate model of the Föppl-von Kármán type from three-dimensional nonlinear elasticity using $Γ$-convergence. By assuming a plate thickness of $h^2$ and a mid-surface with periodic microstructure $h^2\theta(x_1/h, x_2/h)$, it shows that the strain energy scales as $h^8$, leading to a limit model that couples bending ($\nabla^2 v$) and stretching ($\mathop{\rm sym}\nabla{\bf u} + \frac{1}{2}\nabla v \otimes \nabla v$) energies, capturing the mechanical behavior of wrinkled thin structures.

ABSTRACT

In this paper we derive, by means of $Γ$-convergence, the periodically wrinkled plate model starting from three dimensional nonlinear elasticity. We assume that the thickness of the plate is $h^2$ and that the mid-surface of the plate is given by $(x_1,x_2) o (x_1,x_2,h^2θ(\per))$, where $θ$ is $[0,1]^2$ periodic function. We also assume that the strain energy of the plate has the order $h^8=(h^2)^4$, which corresponds to the Föppl-von Kármán model in the case of the ordinary plate. The obtained model mixes the bending part of the energy with the stretching part.

Motivation & Objective

• To justify a two-dimensional plate model for periodically wrinkled thin structures derived from three-dimensional nonlinear elasticity.
• To analyze the asymptotic behavior of a plate with thickness $h^2$ and mid-surface deformation $h^2\theta(x_1/h, x_2/h)$, where $\theta$ is $[0,1]^2$-periodic.
• To establish a limit energy functional that couples bending and stretching contributions, generalizing the classical Föppl-von Kármán model to wrinkled configurations.
• To demonstrate that the resulting model depends on the pre-deformation $\theta$, not just its derivatives, highlighting the role of the initial geometric imperfection.

Proposed method

• Apply $\Gamma$-convergence to the three-dimensional nonlinear elasticity energy, scaled by $h^8$, as $h \to 0$.
• Use two-scale convergence to handle the periodic microstructure in the mid-surface, parameterized by $\theta(x_1/h, x_2/h)$.
• Define the reference configuration as a shell with thickness $h^2$ and mid-surface $\hat{\Omega}^h = \Theta^h(\omega \times (-h^2/2, h^2/2))$, where $\Theta^h$ incorporates the periodic shape function $\theta$.
• Introduce the displacement ansatz $\tilde{\bf y}^h = (\bar{{\bf R}}^h)^T {\bf y}^h - {\bf c}^h$, and decompose the deformation into macroscopic and microscopic components.
• Derive the limiting energy functional $J_0^L({\bf u}, v, \bar{{\bf R}})$, which combines bending ($\nabla^2 v$) and stretching ($\mathop{\rm sym}\nabla{\bf u} + \frac{1}{2}\nabla v \otimes \nabla v$) terms.
• Prove existence of minimizers for the limiting functional in the space $W^{1,2}(\omega;\mathbb{R}^2) \times W^{2,2}(\omega) \times \mathop{\rm SO}(3)$.

Experimental results

Research questions

• RQ1 How does the presence of periodic micro-buckling (wrinkles) in a thin plate affect the effective two-dimensional energy model derived from 3D elasticity?
• RQ2 What scaling of the strain energy ($h^8$) is required to obtain a nontrivial limit model that couples bending and stretching in the wrinkled plate?
• RQ3 Why does the limiting energy functional depend on the pre-deformation $\theta$ itself, not just its derivatives, in contrast to standard plate models?
• RQ4 How does the use of $\Gamma$-convergence and two-scale convergence allow for the derivation of a consistent limit model for a plate with thickness $h^2$ and periodic mid-surface deformation?
• RQ5 What is the role of the rotation $\bar{{\bf R}}$ in the limiting energy, and why does $\bar{{\bf R}}_{33} = \pm 1$ when the load $f_3 \neq 0$?

Key findings

• The limiting energy functional $J_0^L$ couples bending ($\nabla^2 v$) and stretching ($\mathop{\rm sym}\nabla{\bf u} + \frac{1}{2}\nabla v \otimes \nabla v$) terms, reflecting the mechanical interaction between wrinkling and deformation.
• The strain energy scales as $h^8 = (h^2)^4$, which corresponds to the Föppl-von Kármán model order for a plate of thickness $h^2$, justifying the scaling choice.
• The minimizer of the limiting functional exists in the space $W^{1,2}(\omega;\mathbb{R}^2) \times W^{2,2}(\omega) \times \mathop{\rm SO}(3)$, ensuring the mathematical well-posedness of the model.
• For non-zero transverse load $f_3$, the limit rotation satisfies $\bar{{\bf R}}_{33} = 1$ or $-1$, indicating that the plate's orientation stabilizes to a rigid rotation in the limit.
• The model explicitly depends on the shape function $\theta$ itself, not just its derivatives, showing that the pre-deformation geometry is mechanically significant in wrinkled configurations.
• The derivation via $\Gamma$-convergence and two-scale convergence rigorously justifies the model as the $h \to 0$ limit of 3D nonlinear elasticity, establishing its consistency.