Stochastic Homogenization in the Passage from Discrete to Continuous Systems - Fracture in Composite Materials ; Stochastische Homogenisierung im Übergang von Diskreten zu Kontinuierlichen Systemen - Brüche in Verbundwerkstoffen
Titel:
Stochastic Homogenization in the Passage from Discrete to Continuous Systems - Fracture in Composite Materials ; Stochastische Homogenisierung im Übergang von Diskreten zu Kontinuierlichen Systemen - Brüche in Verbundwerkstoffen
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Beschreibung:
The work in this thesis contains three main topics. These are the passage from discrete to continuous models by means of $\Gamma$-convergence, random as well as periodic homogenization and fracture enabled by non-convex Lennard-Jones type interaction potentials. Each of them is discussed in the following. We consider a discrete model given by a one-dimensional chain of particles with randomly distributed interaction potentials. Our interest lies in the continuum limit, which yields the effective behaviour of the system. This limit is achieved as the number of atoms tends to infinity, which corresponds to a vanishing distance between the particles. The starting point of our analysis is an energy functional in a discrete system; its continuum limit is obtained by variational $\Gamma$-convergence. The $\Gamma$-convergence methods are combined with a homogenization process in the framework of ergodic theory, which allows to focus on heterogeneous systems. On the one hand, composite materials or materials with impurities are modelled by a stochastic or periodic distribution of particles or interaction potentials. On the other hand, systems of one species of particles can be considered as random in cases when the orientation of particles matters. Nanomaterials, like chains of atoms, molecules or polymers, are an application of the heterogeneous chains in experimental sciences. A special interest is in fracture in such heterogeneous systems. We consider interaction potentials of Lennard-Jones type. The non-standard growth conditions and the convex-concave structure of the Lennard-Jones type interactions yield mathematical difficulties, but allow for fracture. The interaction potentials are long-range in the sense that their modulus decays slower than exponential. Further, we allow for interactions beyond nearest neighbours, which is also referred to as long-range. The main mathematical issue is to bring together the Lennard-Jones type interactions with ergodic theorems in the limiting process as the number of particles ...