## Towards a material damage model using the logarithmic strain, with von Mises plasticity considerations

Master Thesis

2022

##### Abstract
Damage is briefly defined as the presence and growth of micro-defects in a material. This study serves to describe the computational implementation of the material damage theory adopted for ductile materials. Thus, pays attention to the computational analysis of the physical behaviour of materials under finite deformations — in particular, the stress-strain behaviour, load-deformation behaviour and location of weak zones. Throughout this study, non-linear continuum mechanics is utilised as the mathematical basis of the constitutive and general finite element framework. In continuum mechanics, there exists no requirement to discretely characterise each microcrack that grows in a material, thus making it possible to provide analysis of the stress and strain response affected by micro-defects using material particles, which are localised collections of many atomic-scale particles. The continuum is thus a sum of its material particles. To complement this description of mechanics, constitutive and phenomenological equations are adopted from the non-linear thermodynamic phenomena of elasticity, plasticity, and damage; the laws of thermodynamics will therefore apply and are shown as such. The proposed material damage model is developed and implemented in the backend of the in-house computational mechanics toolbox SESKA, which uses finite element-based discretisation and approximation techniques. Field and scalar quantities, such as stress and strain, are computed with the use of the return-mapping method. The stress measures utilised are the 2nd Piola-Kirchhoff stress S and the Mandel stress Σ. The Newton-Raphson update scheme is applied in the plasticity evolution equations via the plastic multiplier (denoted λ), which innately controls the evolution of all other inelastic phenomena. Damage is a function of plastic evolution and thus plays a role in the plasticity multiplier calculation. Moreover, this proposed model makes the assumption of full isotropy, all material properties at a material point are the same in tension and compression and the same regardless of the dimension. Finally, several examples are utilised to showcase the model and all the intricacies are presented — the problem setup, boundary condition assignment and multi-layered analysis are detailed in the content of this study and the examples perform well under qualitative scrutiny. These examples include a cantilevered beam model, a simply supported bending model and a plane strain example to evaluate whether the material model achieves qualifiable correlation to expected behaviour and to assess whether the damage-related parameters affect the stress and strain behaviour as expected. In brief conclusion, this paper shows that the model achieves qualifiable correlation and all the material parameters function as expected.