Prestrained plates

Prestrained thin materials develop internal stresses, deform out of plane, and exhibit nontrivial 3D shapes even without external forces. Chief examples exhibiting such phenomena are nematic glasses, natural growth of soft tissues or manufactured polymer gels. After reduction from 3D to 2D, denoting by $\Omega\subset\mathbb{R}^2$ the midplane of the plate, the model consists in minimizing the bending energy $$ E(\mathbf{y}) = \frac{\mu}{12}\int_{\Omega}\big|g^{-\frac{1}{2}} \, {\rm II}(\mathbf{y}) \, g^{-\frac{1}{2}}\big|^2+\frac{\lambda}{2\mu+\lambda}{\rm tr}\big(g^{-\frac{1}{2}}\, {\rm II}(\mathbf{y}) \, g^{-\frac{1}{2}} \big)^2 $$ under the constraint that $$\nabla\mathbf{y}^T\nabla\mathbf{y}=g \quad \mbox{a.e. in } \Omega.$$ Here, ${\rm II}(\mathbf{y})$ denote the second fundamental form of the deformed plate $\mathbf{y}(\Omega)$, $\mu$ and $\lambda$ are the Lamé coefficients, and $g$ is the given (symmetric positive definite) target metric. The second fundamental form can actually be replaced by the Hessian of the deformation without affecting the minimizers. In the proposed numerical method, an LDG approach is used for the discretization, where the Hessian is replaced by a so-called discrete Hessian consisting of the broken Hessian and the liftings of jump terms, and an $H^2$ gradient flow is used for the energy minimization.

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Manufactured gel discs

A laboratory experiment, where prestrained discs made of NIPA gel with various monomer concentrations are manufactured, is presented in [3]. Considering a similar metric, see [3, Section 5.5] for details, the proposed numerical method produces deformations that are comparable to the experimental ones.

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Metric with positive Gaussian curvature

In this example, we consider the bending of the unit disc when a metric with positive Gaussian curvature is prescribed, namely $$ g = \nabla\mathbf{y}^T\nabla\mathbf{y} \qquad \mbox{with} \qquad \mathbf{y}(x_1,x_2)=\left(x_1,x_2,\sqrt{0.2}\sin\Big(\frac{\pi}{2}\big(1-\sqrt{x_1^2+x_2^2}\big)\Big)\right)^T, $$ where $\mathbf{y}$ corresponds to the parametrization of a bubble. We do not impose any boundary conditions (free boundary case). Before running the gradient flow algorithm, a pre-processing step is used to get an initial deformation with a prestrain defect below 0.1.

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Metric with negative Gaussian curvature

We use the same setting as in the previous example but with the metric $$ g(x_1,x_2) = \begin{bmatrix} 1+x_2^2 & x_1x_2 \\ x_1x_2 & 1+x_1^2 \end{bmatrix} $$ which has a negative Gaussian curvature and is achieved by the deformation $\mathbf{y}(x_1,x_2)=(x_1,x_2,x_1x_2)^T$ corresponding to a hyperbolic paraboloid.

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Metric corresponding to a catenoid/helicoid

Here, $\Omega\subset\mathbb{R}^2$ is a rectangular domain and we prescribe the metric $$ g(x_1,x_2) = \begin{bmatrix} \cosh(x_2)^2 & 0 \\ 0 & \cosh(x_2)^2 \end{bmatrix}. $$ The deformations $$ \mathbf{y}^{\alpha}(x_1,x_2)=\cos(\alpha) \begin{bmatrix} \sinh(x_2)\sin(x_1) \\ -\sinh(x_2)\cos(x_1) \\ x_1 \end{bmatrix} + \sin(\alpha) \begin{bmatrix} \cosh(x_2)\cos(x_1) \\ \cosh(x_2)\sin(x_1) \\ x_2 \end{bmatrix} $$ are all compatible with the metric and all give the same energy. The case $\alpha=0$ corresponds to an helicoid while $\alpha=\pi/2$ represents a catenoid.