A mathematical model for plasticity and damage: A discrete calculus formulation

TL;DR

This paper introduces a discrete lattice model using discrete calculus to simulate elastic, plastic, and damage behaviors in isotropic materials, enabling detailed analysis of micro-crack growth and material degradation.

Contribution

It presents a novel matrix formulation for a non-linear lattice model incorporating plasticity and damage, facilitating explicit micro-crack analysis in deforming metals.

Findings

Effective simulation of micro-crack generation

Model captures stiffness degradation due to damage

Framework supports explicit analysis of material failure

Abstract

In this article we propose a discrete lattice model to simulate the elastic, plastic and failure behaviour of isotropic materials. Focus is given on the mathematical derivation of the lattice elements, nodes and edges, in the presence of plastic deformations and damage, i.e. stiffness degradation. By using discrete calculus and introducing non-local potential for plasticity, a force-based approach, we provide a matrix formulation necessary for software implementation. The output is a non-linear system with allowance for elasticity, plasticity and damage in lattices. This is the key tool for explicit analysis of micro-crack generation and population growth in plastically deforming metals, leading to macroscopic degradation of their mechanical properties and fitness for service. An illustrative example, analysing a local region of a node, is given to demonstrate the system performance.