The Voronoi cell finite element method for response and damage analysis of arbitrary heterogeneous media

In this dissertation, the Voronoi Cell Finite Element Method (VCFEM) is innovated for material/failure characterization of ductile arbitrary heterogeneous media. The finite element mesh emerges naturally through the Dirichlet tessellation of the microstructure as a network of multisided polygons, with each polygon containing one heterogeneity at most. The resulting Voronoi polygons are treated as finite elements without any further discretization of the microstructural domain. A computational scheme is formulated through the assumed stress hybrid variational principle to solve for the microstructural deformations under the application of macroscopic loading. The two field variational principle automatically allows for multi sided finite elements, through independent assumptions on the deformations on the element boundary and stresses within the element. Constitutive response of the matrix and heterogeneity phases of the microstructure are modeled as rate-independent elastic-plastic {dollar}Jsb2{dollar} materials and the kinematics is analyzed within the framework of 2-d small deformations of the constituents. Damage in the microstructure is initiated by cracking/splitting of the particles triggered through a maximum principle stress theory or Rankine criterion. This model allows for progressive failure of the material without any user interference. Numerical validation of the VCFEM is established through comparison with analytical and traditional finite element methods. Applicability of the current scheme to model damage in arbitrary media is demonstrated for computer generated/actual microstructures. Effect of stereographic features such as shape, size and distribution of heterogeneities on damage evolution in random microstructures are also discussed. Statistical representations, that correlate microstructural morphology with material failure, are used to derive the required length scales for the microstructural problem.