Uniformly Gamma-equivalent theories for discrete-to-continuum limits
Final Report Abstract
In semiconductor optoelectronics and other areas of engineering there is a strong interest in small devices on the nanoscale or microscale. Standard mathematical models that describe the mechanical properties of such devices reach their limit of applicability. On the other hand, fully atomistic models are often too complex or time-consuming for numerical simulations. This is why we mathematically analyze the influence of microscale structures on the overall effective behaviour of a material. To this end we start from a model in the atomistic/discrete setting and study its behaviour as the number of atoms tends to infinity or the distance between neighbouring atoms tends to zero. This demands appropriate scalings of the energy functionals and reasonable mathematical tools which recover important physical properties of the microscopic systems. For instance, minimizers of the atomistic system have to converge to the minimizers of the limiting problem, which is ensured in the context of so-called Γ-convergence methods we apply. In this project we in particular gained a deeper insight into the influence of cracks and dislocations on the macroscopic behaviour of the specimens. We rigorously derived formulas for surface energies that are based on the microscopic parameters of the system. That is, instead of assuming the surface energies ad-hoc, one can calculate the surface energy just based on underlying microscopic quantities. Further, in the context of a so-called quasicontinuum method, which is a computational tool that is frequently used in engineering to simulate specimens with cracks or dislocations, we derived (in a one-dimensional setting) a condition on the choice of the numerical mesh, which ensures that the minimal energy of the asymptotic quasicontinuum model is the same as the minimum of the asymptotic energy formula of the fully atomistic model. Finally, we rigorously verified that in heterogeneous nanowires dislocations occur if the radius of the wire becomes too large. Mathematically this involved a passage from discrete to continuous systems as well as a dimension reduction from three-dimensional discrete system to a one-dimensional continuum system approximating the nanowire. A deeper understanding of the formation of dislocations in such nanowires is in particular interesting for applications of these wires in semiconductor optoelectronics since dislocations change the electronic properties.
Publications
- On Lennard-Jones systems with finite range interactions and their asymptotic analysis, 33 pages
M. Schäffner and A. Schlömerkemper
- Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires, 21 pages
G. Lazzaroni, M. Palombaro and A. Schlömerkemper
- Boundary layer energies for nonconvex discrete systems, Math. Models Methods Appl. Sci. (M3AS) 21, 777–817 (2011)
L. Scardia, A. Schlömerkemper and C. Zanini
- Towards uniformly Γ-equivalent theories for nonconvex discrete systems, Discrete Contin. Dyn. Syst. B 17, 661–686 (2012)
L. Scardia, A. Schlömerkemper and C. Zanini
(See online at https://doi.org/10.3934/dcdsb.2012.17.661) - About an analytical verification of quasi-continuum methods with Γ-convergence techniques, Mater. Res. Soc. Symp. Proc. 1535 (2013)
M. Schäffner and A. Schlömerkemper
(See online at https://doi.org/10.1557/opl.2013.458) - Dislocations in nanowire heterostructures: from discrete to continuum, PAMM Proc. Appl. Math. Mech. 13, 541–544 (2013)
G. Lazzaroni, M. Palombaro and A. Schlömerkemper
- A discrete to continuum analysis of dislocations in nanowire heterostructures, Comm. Math. Sci. (2015)
G. Lazzaroni, M. Palombaro and A. Schlömerkemper
- On a Γ-convergence analysis of a quasicontinuum method, Multiscale Model. Simul. 13, 132–172 (2015)
M. Schäffner and A. Schlömerkemper
