# Stiffness and Density Relationships in Additively Manufactured Structures: A Virial Theorem-Based Approach

## TL;DR

This paper introduces a new mathematical model based on the virial theorem to predict stiffness-density relationships in 3D-printed structures.

## Contribution

A novel virial theorem-based framework for predicting stiffness-density relationships in additively manufactured structures is introduced.

## Key findings

- The virial theorem provides a predictive framework for stiffness-density relationships in optimized structures.
- Optimal density in structures is linked to the golden ratio.
- The model is generalizable to broader materials science and manufacturing applications.

## Abstract

Topological optimization uses two main optimization conditions aimed at achieving the maximum stiffness at minimum weight of the loaded object, while not exceeding the allowable stress. This process naturally creates complex structures with varying degrees of density. There is a certain regularity between the density of the structure and stiffness, with the optimal density being related to the golden ratio. This study contributes to materials modeling and their characterization by introducing a mathematical theory related to the virial theorem as a predictive framework for understanding stiffness–density relationships in additively manufactured structures. The definition of virial stability and the methodology for deriving this stability from the kinetic and potential components of a random signal are introduced. The proposed virial-based model offers a generalizable tool for materials characterization, applicable not only to topological optimization but also to broader areas of materials science and advanced manufacturing.