# Printing non-Euclidean solids

## TL;DR

This paper explores how incompatible surface growth can be used to control the behavior of non-Euclidean solids, with applications in 3D printing arteries and explosive plants, by tuning geometrical frustration during deposition.

## Contribution

It introduces a mechanics framework for incompatible surface growth to achieve targeted configurations in non-Euclidean solids, including explicit protocols for biological and technological applications.

## Key findings

- Stress uniformity in arteries under inhomogeneous loads
- Complete energy release in explosive plants with a single cut
- Topological incompatibility is essential for physiological targets

## Abstract

Geometrically frustrated solids with non-Euclidean reference metric are ubiquitous in biology and are becoming increasingly relevant in technological applications. Often they acquire a targeted con- figuration of incompatibility through surface accretion of mass as in tree growth or dam construction. We use the mechanics of incompatible surface growth to show that geometrical frustration develop- ing during deposition can be fine-tuned to ensure a particular behavior of the system in physiological (or working) conditions. As an illustration, we obtain an explicit 3D printing protocol for arteries, which guarantees stress uniformity under inhomogeneous loading, and for explosive plants, allowing a complete release of residual elastic energy with a single cut. Interestingly, in both cases reaching the physiological target requires the incompatibility to have a topological (global) component.