Christof Eck's research while affiliated with Universität Stuttgart and other places

... Models play a crucial role in the scientific process by providing a representation of complex systems, processes, and phenomena. Models help scientists to make predictions, test hypotheses, and gain a deeper understanding of the behavior of these systems [1,2]. By using mathematical models, such as physical, statistical, or simulation models, scientists can study the relationships between variables, estimate uncertainties, and explore scenarios without having to perform expensive or dangerous experiments [3,Ch. ...

... Moreover, it turns out that, apart from differences in boundary conditions and source terms, the mathematical formu- lation of the problem is more or less identical with that of a number of problems in porous medium flow [2], [3] (actually, our problem is a problem in porous medium flow), Hele Shaw flows [9], [10], [14], electrochemical machining [8] and heat conduction (degenerate one-phase Stefan problems) [6], [7], [1]. Since the -2- literature on these problems is now fairly extensive and easily available (e.g. in [ 1 0] , [12], [16] f [24]) there is no need to give here an exhaustive treatment of the problem. ...

... is based on continuum mechanical modelling concepts and comprises two nonlinear equations derived from mass balance considerations and the Reynolds transport theorem, [45,Satz 5.4]; Equation (1.1) models the wetting fluid phase and (1.2) models the nonwetting phase. We briefly sum up the meaning of the appearing entities and touch on a few modelling concepts. ...

... Therefore, currently, continuum mechanical simulations enable engineers to simulate the behavior of solids on much larger time and length scales than employing molecular dynamical simulations. In the present thesis, we aim at the (dynamic) continuum mechanical simulation of a solid's reaction to large external forces basically following the monographs [Cia88,EGK08]. Therefore, in the presented model, all physical quantities, for instance, mass, linear momentum, velocity and energy, are considered as mean values. ...

... Farah et al. [16] proposed a new hybrid scheme which beneficially combines the advantages of segment-based and element-based mortar integration. These contact constraint methods have been combined with discrete numerical methods, such as boundary element method [17][18][19], meshless method [20,21], and finite element method [3,4]. With the application of the finite element method (FEM), the FEM has occurred frequently in contact analysis in the past decades [22][23][24]. ...

... are reaction rates according to mass action law, cf. [3, 14, 45, 53, 55]. More precisely, from these references we recall that provided the stoichiometry of a chemical reaction reads as˜s ...

... The dependence of this approximation of the sharp interface velocity on the choice of µ was only clarified in a further publication [2]. Similar approaches can be found in [76], [43] and [34]. ...

... The physical models describing phase transformations, with or without mechano-chemical coupling, can be classified into two main approaches: i) the two-field approaches [31][32][33][34][35][36][37], in which the mass transport is resolved within each phase separately, with a condition at the interface for mass conservation which induces the interface movement; ii) the one-field approaches in which mass transport and transfer are modelled by a unique equation for the overall system [38][39][40]. Phase field modelling belong to the latter since it is based on a free energy functional defined at every point of the multi-phase systems [41][42][43][44][45][46]. They present great advantages for modelling the evolution of complex 3D microstructures. ...

... Although phase field models are widely used and have many applications, there exists only little work on them in homogenization settings. The earliest work known to the author is on a phase field model for the liquid-solid phase transition of binary mixtures ( [Eck and Knabner, 2002], [Eck, 2004], and [Eck, 2005]). The phase field model under consideration is a generalized Caginalp phase field model and describes the dendritic evolution of microscopic crystals. ...

... We now describe the physical setting of the contact problem and provide its classical formulation. We follow the exposition of the model described in [1,8,23,[36][37][38] and employ the thermoviscoelastic version of the Kelvin-Voigt constitutive law. Thus, the thermomechanical behavior of the material is assumed to be linear while the nonlinear effects occur only in the contact boundary conditions. ...