3D Periodic Cellular Materials with Tailored Symmetry and Implicit Grading

Highlights

• 3D periodic cellular materials are an essential ingredient in additive manufacturing.

• We study a novel type of 3D periodic materials that emerge from a growth process.

• Our approach enables restricting the symmetry of the elastic responses.

• We provide numerical and experimental results of the structures’ elastic properties.

Abstract

Periodic cellular materials allow triggering complex elastic behaviors within the volume of a part. In this work, we study a novel type of 3D periodic cellular materials that emerge from a growth process in a lattice. The growth is parameterized by a 3D star-shaped set at each lattice point, defining the geometry that will appear around it. Individual tiles may be computed and used in a periodic lattice, or a global structure may be produced under spatial gradations, changing the parametric star-shaped set at each lattice location.

Beyond free spatial gradation, an important advantage of our approach is that elastic symmetries can easily be enforced. We show how shared symmetries between the lattice and the star-shaped set directly translate into symmetries of the periodic structures’ elastic response. Thus, our approach enables restricting the symmetry of the elastic responses – monoclinic, orthorhombic, trigonal, and so on – while freely exploring a wide space of possible geometries and topologies. We make a comprehensive study of the space of symmetries and broad combinations that our method spans and demonstrate through numerical and experimental results the elastic responses triggered by our structures.

Introduction

3D periodic cellular materials are an essential ingredient in the manufacture of advanced functional parts. They allow triggering various elastic responses within a part by structuring its internal volume at a small scale. Ideally, a method to model cellular materials would allow for a wide gamut of elastic responses and geometries that could be freely chosen within the part volume.

A central line of research is the design of periodic cellular materials, defined from a representative volume element (or tile) repeated in space. The periodicity simplifies the design and analysis while limiting computational and storage requirements. We consider two types of cellular materials [1]: closed-cell structures which are formed by isolated void cavities (cells), and open-cell structures, formed by connected solid beams (possibly curved).

A key to achieving complex, dynamic behaviors [2] is to allow for free spatial gradation of the structures within the part volume. Most techniques employ a catalog of tiles, pre-determined or optimized, placing different tiles with different elastic responses in different locations. Adjacent tiles will generally not connect properly, requiring applying constraints on the sets of tiles or optimizing after placement (see Section 2).

In this work, we take a distinct view on this problem. Rather than reasoning in terms of a fixed number of tiles, we start from a periodic lattice of nuclei (points) and associate each parametric description of the local structure being produced — a 3D star-shaped set. Each star-shaped set implicitly defines a distance function. Since the star-shaped set is possibly non-convex, we span a wide range of possible distance functions and a more diverse space of cellular solids compared to existing techniques. An efficient computational process grows the structures around each nucleus. Intuitively speaking, the cells centered around the lattice nuclei grow, obeying a law of growth governed by the star-shaped distance while being forbidden to overlap. Once the growth process is complete, we obtain a cellular structure, which can be extracted as either closed or open-cell. If the same star-shaped distance is used at every nucleus, we obtain a periodic structure, from which a single period can act as a traditional periodic tile. If different star-shaped distances or lattices are used, we obtain spatial gradations.

Resorting to a classical Voronoi diagram under the star-shaped distance may lead to Voronoi cells with more than one connected component. Instead, the growth process ensures that each cell is connected by construction. More precisely, we consider a discrete growth process since it is challenging to formulate a continuous one: our growth process is atypical, guided by the growth of a non-convex set. To the best of our knowledge, continuous growth processes have been studied for convex distances (e.g. [3]) but not for star-shaped ones.

Our structures can be prescribed to enforce certain crystallographic geometric symmetries — and hence a given type of elastic response: monoclinic, orthorhombic, trigonal, and so on. The parametric star-shaped distances can also be easily interpolated, making it possible to transition from one geometry to another smoothly. Thus, our approach supports interpolation across different symmetries and topologies without any special treatment. It spans a broad space of possible geometries with controlled symmetries that can be freely combined and spatially graded. Finally, the compact parameterization of the distance and the lattice facilitates exploring the wide variety of possible cellular materials.

We now define the terminology used throughout the paper. A tile, or representative volume element of a structure, is a periodic computational domain. The domain is covered by a regular grid, a set of discrete points each representing the center of a voxel (a cubic cell of the grid centered on the point). We define a nucleus as the starting point of growth of cavities or cells on the grid. We call a lattice the discrete set of nuclei distributed within the tile. The lattice has translational symmetries; therefore, it can be represented by periodic crystals [4]. Finally, a compact set $\mathcal{S}\subset {\mathbb{R}}^3$ is star-shaped with respect to the origin $O$ if for all $x\in \mathcal{S}$, the segment $[O,x]$ is contained in $\mathcal{S}$.