Direct coupling of continuum and shell elements in large deformation problems

TL;DR

This work addresses the computational challenge of simulating thin-walled reinforcements embedded in bulk solids under large deformations by introducing a mixed shell discretization that directly couples shell mid-surfaces to volume elements. A geometric and kinematic reduction replaces the shell domain with its mid-surface while deriving a reduced surface energy density that separates membrane and bending effects via $oldsymbol{e}$ and $oldsymbol{ ext{κ}}$, yielding a tractable variational problem. To enable standard nodal discretizations across the coupled domains, the authors develop a low-regularity, three-field (and hybridized) shell formulation that avoids demanding $C^1$ continuity and can handle kinks and branched interfaces; local equilibrium is shown to be consistent in the linear setting. Computational results on a square laminate block, a submerged half-cylinder, and a wrinkling PDMS-like problem demonstrate high accuracy and dramatic reductions in degrees of freedom, pointing to substantial practical impact for complex reinforced structures in soft matrices.

Abstract

In many applications, thin shell-like structures are integrated within or attached to volumetric bodies. This includes reinforcements placed in soft matrix material in lightweight structure design, or hollow structures that are partially or completely filled. Finite element simulations of such setups are highly challenging. A brute force discretization of structural as well as volumetric parts using well-shaped three-dimensional elements may be accurate, but leads to problems of enormous computational complexity even for simple models. One desired alternative is the use of shell elements for thin-walled parts, as such a discretization greatly alleviates size restrictions on the underlying finite element mesh. However, the coupling of different formulations within a single framework is often not straightforward and may lead to locking if not done carefully. Neunteufel and Schöberl proposed a mixed shell element where, apart from displacements of the center surface, bending moments are used as independent unknowns. These elements were not only shown to be locking free and highly accurate in large-deformation regime, but also do not require differentiability of the shell surface. They can directly be coupled to classical volume elements of arbitrary order by sharing displacement degrees of freedom at the center surface, thus achieving the desired coupled discretization. As the elements can be used on unstructured meshes, adaptive mesh refinement based on local stress and bending moments can be used. We present computational results that confirm exceptional accuracy for problems where thin-walled structures are embedded as reinforcements within soft matrix material.

Figures (11)

• Figure 1: Geometric reduction of the shell domain $\Omega_\mathcal{S}$ to a representation through its mid-surface $\mathcal{S}$. Top: The shell domain is embedded into the bulk domain. Bottom: The shell domain is located at the border of the bulk domain.
• Figure 2: Normal and tangential vectors in the finite element mesh. Left: surface normals. Right: in-plane edge normals.
• Figure 3: Square laminate plate: geometric setup of the symmetric block with kinematically constrained surfaces indicated in dark gray (left); different variants of through-the-thickness position of reinforcements (right).
• Figure 4: Square laminate plate: load case extension for $\Delta L/L = 0.1$, reaction force in longitudinal $x$ direction as well as total displacement $\boldsymbol u$ in points $A$ and $B$ are monitored for different stiffness ratios $E_\mathcal{B}/E_\mathcal{S} \in [0.001, 1]$. Relative errors of shell discretization with respect to full 3d results are displayed; left: through-the-thickness setup with surface reinforcements; right: through-the-thickness setup with immersed reinforcements.
• Figure 5: Square laminate plate: load case simple shear for $\Delta L/L = 0.1$, reaction force in transverse $z$ direction as well as total displacement $\boldsymbol u$ in points $A$ and $B$ are monitored for different stiffness ratios $E_\mathcal{B}/E_\mathcal{S} \in [0.001, 1]$. Relative errors of shell discretization with respect to full 3d results are displayed; left: through-the-thickness setup with surface reinforcements; right: through-the-thickness setup with immersed reinforcements.