Homogenization technique to perform periodic unit cell calculations using spring elements

Abstract

The objective of this thesis is to implement the spring element-based homogenization technique to perform periodic unit cell calculations within the ABAQUS finite element framework. Unit cell calculations are critical in the context of ductile failure analysis, as they provide a detailed understanding of localized deformation mechanisms, such as plastic flow and stress evolution. A mathematical framework is implemented to represent the elastic–plastic behaviour of the matrix material, incorporating isotropic hardening under proportional loading conditions. The governing equations are formulated using finite strain theory and executed through finite element modelling using the ABAQUS/Standard implicit solver. The model utilizes spring elements to enforce periodic boundary conditions and control stress application, enabling simulations under controlled multiaxial stress states. The results show that the accuracy of the unit cell simulation is primarily influenced by two key parameters: the spring stiffness and the size of the time increment. Improper calibration of these parameters can lead to notable deviations from target stress ratios, especially in the plastic regime. Through performance assessment and sensitivity analysis, the study identifies optimal ranges for these parameters, ensuring a balance between numerical accuracy and computational efficiency. This work establishes a robust and adaptable approach to micromechanical simulation, with potential extensions to dynamic loading conditions and porous materials in future research.