Resumen de Discrete volume method. A variational approach for brittle fracture
Victor Eduardo Cardoso Nungaray
This thesis presents a proposal to simulate mechanics and dynamics of brittle fracture. A variational formulation is used to describe Lagrangian mechanics, by minimizing the difference between potential and kinetic energy of the system, obtaining a pair of partial differential equations; the solution of these equations corresponds to the displacement field and damage phase-field respectively. Such an equations are coupled in the sense that the damage field is used in the first equation and the displacement field is used in the second one.
In this work we propose a numerical method based on control volumes to solve the differential equations, extending the formulation to support the separation of control volumes, processing these volumes as discrete entities. This treatment results in accurate calculations of stress field and the nucleation of new internal fractures that can be propagated through domain creating multiple bifurcations.
To integrate equations inside control volumes we introduce a family of polynomial splines that we refer as homeostatic splines, since its derivatives are null at vertices with a smooth function variation between adjacent volumes. Furthermore, we propose a shape function with trigonometric components for dynamic analysis, allowing bigger time steps that with traditional approaches.
Finally, we perform ten numerical experiments to show the effectiveness of the method and to compare our results with those published by other authors.
