Multiscale Modeling of Damage in Micro-Heterogeneous Materials based on incremental variational formulations
Final Report Abstract
During the funded research stay in the group of Professor Michael Ortiz at the California Institute of Technology a relaxed incremental variational formulation for damage at large strains has been developed. The model founds on the notion of continuum damage mechanics and takes into account a standard formulation as long as the incremental stress potential is convex. As soon as convexity is lost, the initially homogeneous material is assumed to bifurcate into a weakly and strongly damaged phase. Then a relaxed stress potential is considered which is mainly obtained by a geometric construction of an interpolation between the two bifurcated states. Thereby, the resulting model can be interpreted as the homogenization of the microheterogeneous distribution of weakly and strongly damaged phases. The main advantage of this method is that existence of minimizers of the variational problem is ensured which plays an important role with respect to numerical calculations. Since in various materials fiberor truss-like elements occur at the microscale that undergo damage during deformation, the consideration of a one-dimensional relaxed damage formulation for the directional elements turned out to be efficient. Therefore, the special one-dimensional case has been derived and a model for fiber-reinforced materials was proposed. This model basically takes into account a local transformation of the deformations into the fiber directions, into the evaluation of the relaxed damage model, and then into transforming the stresses and moduli back to the three-dimensional space. The developed model was implemented in the Finite-Element-Analysis-Program (FEAP by Robert Taylor from the University of California in Berkely) and several numerical examples were calculated. First, mesh-independency of the one-dimensional model was shown by analyzing different meshes of a tension test of a distorted beam. Second, a multiscale application was investigated, where the numerical homogenization of a microtruss material was considered. There, it was shown that the unrelaxed formulation suffers from convergency problems in individual microtrusses even for a high number of time steps, whereas the relaxed formulation leads to converging results at a reasonable number of time steps. Finally, a fiber-reinforced cantilever beam was analyzed showing the influence of the anisotropy due to the fiber-reinforcement leading to an unsymmetric deformation. At Caltech it was a great experience to meet high potential researchers from various fields of engineering and discuss topics going beyond the notion of mechanics known so far to the applicant. This led to an additional highly interesting project which was not planned in the original funding application. The idea came up to bring together computational multiscale simulation of DP-steel, what the applicant has been working on comprehensively in the last years and “Optimal Uncertainty Quantification”, a method developed at Caltech by Tim Sullivan (Graduate Laboratories in Aerospace), Houman Owhadi (Applied & Computational Mathematics and Control & Dynamical Systems) and Michael Ortiz (Graduate Laboratories in Aerospace and Mechanical Engineering). Thereby, the quantification of optimal bounds of uncertainties occurring in multi-phase steels was planned to be calculated leading to complex minimization/maximization problems, which were only solvable in a strong cooperation with Mike McKerns (Caltech Center for Advanced Computing Research), who has been modifying his optimization code. This environment is specialized for the problems arising in uncertainty quantification. My part has been the enhancement of my multiscale code for the simulation of DP-steels such that an automated calculation is enabled. This is essential for the solution of the optimization problems required by the “Optimal Uncertainty Quantification” approach. In addition to that, an interface was developed which enables the two computing frameworks to communicate and to still enable the optimizing framework to run in parallel. Summarizing, the DFG-fellowship enabled the applicant to take a close look into the field of incremental variational formulations and relaxation and contribute scientifically to these fields by developing a new damage approach. This was only possible in a close cooperation with Professor M. Ortiz which is planned to be continued in the future. In addition to that, a further cooperation with respect to another scientific topic of high relevance, the uncertainty quantification, was established.
Publications
- Graduate Laboratories in Aerospace, California Institute of Technology, December 3, 2010: “Application of optimal uncertainty quantification to the certification of multiphase alloy material response
D. Balzani, T. Sullivan, M. McKern, and M. Ortiz
- Graduate Laboratories in Aerospace, California Institute of Technology, October 15, 2010: “Coupled micro-macro simulation of multiphase steels based on statistically similar RVEs”
D. Balzani
- Caltech Solid Mechanics Symposium, January 19, 2011, California Institute of Technology, Pasadena: “Optimal uncertainty quantification and certification of material response”
T. Sullivan, D. Balzani, M. McKerns, and M. Ortiz
- Department of Mechanical Engineering at Stanford University, February 17, 2011, 2011: “Simulation of atherosclerotic arteries - generalized convexity conditions, modeling of damage and residual stresses”
D. Balzani
