We model the formation of wrinkles of an elastic substrate coated with a thin film. The elastic substrate is first stretched, then the film is attached to a part of the substrate boundary in the deformed state. Once the the external force is released, wrinkles form due to the stress mismatch between the two materials. The elastic substrate is modeled using a hyperelastic, homogeneous and isotropic material. The film is modeled using a geometrically exact Cosserat shell. The resulting deformation and microrotation $(varphi, R)$ are a minimizing pair of the combined energy functional
$$
J(varphi, R) = int_{Omega} W_textup{bulk}(nablavarphi) : dV + int_{Gamma_c}W_textup{coss}(nablavarphi_{|_{Gamma_c}}, R) : dS
$$
in the admissible set
begin{align*}
mathcal{A} = Big{&(varphi, R) in W^{1,q}(Omega, mathbb{R}^3) times H^1(Gamma_c,textup{SO(3)}) : Big| :
varphi textnormal{ is a deformation function, }& (varphi, R) textnormal{ fullfill the Dirichlet boundary conditions} Big}
end{align*}
with $q > 3$.
We discretize the problem using Lagrange finite elements for the substrate displacement. For the numerical treatment of the microrotation field, standard Lagrange finite elements cannot be used, as the microrotation field maps to the nonlinear manifold $textnormal{SO}(3)$. We present a generalization of Lagrange finite elements that is suitable for such manifold-valued functions: geometric finite elements.
The resulting finite element spaces are complete and invariant under isometries of the manifold. The best approximation error depends on the mesh size h. We prove the existence of solutions of the discrete coupled model. We compare two Newton-type methods to solve the resulting discrete problem: a Riemannian trust-region method and a Riemannian proximal Newton method.
Numerical experiments show that we can efficiently reproduce wrinkling patterns of coupled systems. Our approach works as well for more complex scenarios like multi-layer systems or systems involving various stress-free configurations.