Mathematical Modeling of Networks in Strength of Materials

TL;DR

The paper addresses modeling a material as a discrete node-edge network and derives a nonlinear algebraic system that links external forces, internal bond forces, and nodal positions. A discrete framework is developed using the incidence matrix $A$ and a bond-force law $F = K(y)\, y$, where $K(y)$ is diagonal and depends on deformation, leading to the core equation $A^T K(AX) AX = B$. Unknowns are the free node displacements $\mathbf{X}_P$ and their associated reactions $\mathbf{B}_P$, with $\mathbf{X}_Q$ and $\mathbf{B}_Q$ fixed by boundary conditions. The paper outlines a two-step path to publishable results: (i) solve the nonlinear system numerically and (ii) validate with concrete 3D configurations (triangle and octahedron) to demonstrate elastic, plastic, and potential cracking behavior in the network.

Abstract

We study a material modeled as a network of nodes connected by edges. Using a discrete approach, we build a nonlinear algebraic system that connects applied forces to internal forces and node positions. The model can describe elasticity, plasticity, and possibly cracking. The goal is to solve this system and understand how the material responds. Students are asked to start with a simple triangle example and then apply the method to larger structures. The final aim is to solve the full system and justify the results, leading to a possible publication.

Figures (1)

• Figure 1: Graph showing the relationship between the force magnitude $|F_i|$ and the extension $|y_i| - |b_i|$ of a bond.