On the Formulation and the Micromechanical Origin of Non-Classical Models of Diffusion
Applied Mechanics, Statics and Dynamics
Final Report Abstract
In this research project, we have studied classical first gradient-type diffusion, e.g of Fickian type, and nonclassical second gradient-type diffusion, e.g. of Cahn–Hilliard type. These models are used to describe transport of a mobile species of atoms through rigid and elastic solids. Second gradient-type diffusion even supports the formation of distinct solid phases, characterized by high and low concentrations of the mobile species, respectively. Particular attention has been paid to modeling of diffusion at solid-solid interfaces in particulate composites. Starting from the underlying principles of chemomechanics we have established a numerical framework by linearizing and discretizing the underlying set of equations. Various discretization methods have been implemented and compared to each other. A two-scale formulation of the coupled problem has been established. The main results of our work can be summarized as: • Implementation of first and second gradient-type diffusion in rigid solids using the isogeometric analysis (IGA), the natural element method (NEM), and the finite element method (FEM). The former two are capable of providing C1-continuous approximations of the test and solution fields which are required for a direct modeling approach of second gradient-type models. A second-order splitting approach is used together with standard C0-continuous finite elements. There, the concentration of the mobile species and the corresponding chemical potential are introduced as independent variables. As an alternative to this, a micromorphic model, which introduces an additional non-local variable next to the concentration, has also been implemented using standard finite elements. The implementation using IGA and the second-order splitting FEM were found superior to the other approaches. • Thermodynamically consistent modeling of chemomechanical solid-solid interfaces. While being mechanically perfect, the considered interface is chemically imperfect. The mobile species flux across the interface is limited in terms of the chemical potential jump, as it is suggested by the dissipation inequality. In addition to that the interface is assumed to be energetic, that is it is equipped with its own energy, entropy, balance equations, and constitutive relations. After formulation of chemical and mechanical balances, thermodynamically consistent constitutive relations have been derived. The used interface energy depends on the concentration as well on the deformation and has a significant influence of the overall response of the body. In particular, it allows that the diffusive interface formed during phase separation is not necessarily perpendicular to the interface. • The constitutive response of a generic class of macro-homogeneous diffusors was modeled using numerical first-order homogenization of micro-heterogeneous diffusors with energetic imperfect interfaces. Tho this end, a variationally consistent two-scale modeling approach has been utilized. Adequate relations for the bridging between both scales, the macro and the microscale, have been derived. Size effects are introduced to the macroscopic response due to the finite size of the used representative volume element and, in addition to that, due to the presence of the energetic interface. Overall, the response of the macro-homogeneous body obtained by two-scale formulation is in good agreement with the response of the corresponding micro-heterogeneous body obtained by direct numerical simulation for a wide range of parameters. The respective six publications in peer-reviewed journals and the PhD thesis have been well received as indicated by almost 100 citations (12.2021).
Publications
- A study on mixed finite element formulations applied to diffusion problems, Proceedings in Applied Mathematics and Mechanics, 14(1):485–486 (2014)
Kaessmair, S., Javili, A., amd Steinmann, P.
(See online at https://doi.org/10.1002/pamm.201410230) - General imperfect interfaces, Computer Methods in Applied Mechanics and Engineering, 275:76–97 (2014)
Javili, A., Kaessmair, S., and Steinmann, P.
(See online at https://doi.org/10.1016/j.cma.2014.02.022) - Thermomechanics of solids with general imperfect coherent interfaces, Archive of Applied Mechanics, 84(9-11):1409–1426 (2014)
Kaessmair, S., Javili, A., amd Steinmann, P.
(See online at https://doi.org/10.1007/s00419-014-0870-x) - Comparative computational analysis of the Cahn–Hilliard equation with emphasis on C1 –continuous methods. Journal of Computational Physics 322: 783-803 (2016)
Kaessmair, S. and Steinmann, P.
(See online at https://doi.org/10.1007/s00419-014-0870-x) - On the computational homogenization of transient diffusion problems, Proceedings in Applied Mathematics and Mechanics, 16(1): 529–530 (2016)
Kaessmair, S., Steinmann, P.
(See online at https://doi.org/10.1002/pamm.201610253) - Computational first-order homogenization in chemo-mechanics. Archive of Applied Mechanics 88: 271–286 (2018)
Kaessmair, S. and Steinmann, P.
(See online at https://doi.org/10.1007/s00419-017-1287-0) - Computational mechanics of generalized continua. In: Altenbach, H. and Öchsner, A (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg (2020)
Kaessmair, S. and Steinmann, P.
(See online at https://doi.org/10.1007/978-3-662-55771-6_111) - Multi-scale modelling of diffusion in elastic composite materials. Ph.D thesis, Universität Erlangen-Nürnberg (Schriftenreihe Technische Mechanik, 36), Erlangen (2020)
Kaessmair, S.
- Variationally Consistent Computational Homogenization of Chemo-Mechanical Problems with Stabilized Weakly Periodic Boundary Conditions. International Journal of Numerical Methods in Engineering 122: 6429—6454 (2021)
Kaessmair, S., Runesson, K., Janicke, R., Steinmann, P., and Larsson, F.
(See online at https://doi.org/10.1002/nme.6798)
