Multiscale Modeling and Simulation of Damage by Void Nucleation and Growth

Voids are observed to be generated under sufficient loading in many materials, ranging from polymers and metals to biological tissues. The presence of these voids can have drastic implications at the macroscopic level including strong material softening and more incipient fracture. Developing tools to appropriately account for these effects is therefore very desirable. This thesis is concerned with both, the appearance of voids (nucleation process) and the modeling and simulation of materials in the presence of voids. A particular nucleation mechanism based on vacancy aggregation in high purity metallic single crystals is analyzed. A multiscale model is developed in order to obtain an approximate value of the time required for vacancies to form sufficiently large clusters for further growth by plastic deformation. It is based on quantum mechanical results, kinetic Monte Carlo methods and continuum mechanics estimates calibrated with quasi-continuum results. The ultimate goal of these simulations is to determine the feasibility of this nucleation mechanism under shock loading conditions, where the temperature and tensions are high and vacancy diffusion is promoted. On the other hand, the effective behavior of materials with pre-existent voids is analyzed within the general framework of continuum mechanics and is therefore applicable to any material. The overall properties of the heterogeneous material are obtained through a two-level characterization: a representative volume element consisting of a hollow sphere is used to describe the “microscopic” fields, and an equivalent homogeneous material is used for the “macroscopic” behavior. A variational formulation of this two-scale model is presented. It provides a consistent definition of the macro-variables under general loading conditions, extending the well-known static averaging results so as to include microdynamic effects under finite deformations. This variational framework also provides a suitable starting point for time discretization and consistent definitions within discrete vii time. The spatial boundary value problem resulting from this multiscale model is solved with a particular spherical shell element specially developed for this problem. The approximation space is based on spherical harmonics, which respects the symmetries of the porous material and allows the representation of the fields on the sphere with very few degrees of freedom. Numerical tools, such as the exact representation of the boundary conditions and an exact quadrature rule, are also provided. The resulting numerical model is verified extensively, demonstrating good convergence results, and its applicability is shown through several material point calculations and a full two-scale finite element implementation.