Browsing by Author "Sändig, Anna-Margarete (Apl. Prof. Dr.)"

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Open Access
The Cahn-Larché system : a model for spinodal decomposition in eutectic solder ; modelling, analysis and simulation
(2005) Merkle, Thomas; Sändig, Anna-Margarete (Apl. Prof. Dr.)
Electronic control of mechanical procedures in particular within an automobile becomes recently more and more important. Due to this fact reliability and life time of a solder joint in a control device become significant for automotive industry. Experimental investigations of a solder joint where the configuration is subjected to several thousand power cycles show an essential change of the microstructure in the alloy. The originally fine mixture separates into two or more phases. However, regions with a coarse microstructure are not randomly distributed over the solder joint. They are located in a vicinity of a notch or a reentrant corner or lie nearby a hard clamped boundary part of the solder bump. In the worst case cracks appear in the alloy between regions of coarse and fine microstructure. The phase separation process is modelled by a diffusive phase interface model, which was derived by Cahn and Hilliard and extended by Cahn and Larché in order to consider elastic effects. In this thesis a mathematical rigorous derivation of the Cahn-Larché system is presented, which additionally takes into account external mechanical loadings and viscosity. The examination of the general entropy principle is done by applying Lagrangian multipliers to the thermodynamics of the spinodal decomposition. The existence of a weak solution of the viscous Cahn-Larché system is shown under consideration of a concentration dependent mobility tensor and external mechanical forces. In order to attain this result, we extend the method developed by Garcke consisting of a time discretisation, minimising the internal energy and maximising the dissipation. An interesting result of our investigations is the fact that the a-priori estimates do not depend on the friction coefficient. Due to this observation we simultaneously get the existence of a weak solution of the viscous and non-viscous system. Thereby we observe that a weak solution of the viscous system is smoother with respect to time than a solution of the non-viscous system. The numerical simulations of the phase separation are done by using a Faedo-Galerkin method. Extremely small surface stress tensors cause the Gibbs phenomena. This means that high overshoots and undershoots appear within a diffusive interface. In order to solve this problem a numerical approximation method stabilised by dynamical friction is developed. A second viscous approximation method is analyzed, where friction is formulated in terms of driving forces. We show the equivalence of both methods by using the flow gradient structure of the system. Finally, an operator-splitting method is derived, where the Gibbs free energy density is decomposed into a convex and a concave part. The different numerical simulations show that the Cahn-Larché system with a concentration dependent elasticity tensor fits qualitatively better to the experiment than the simple model with a constant elasticity tensor. Mechanical stress singularities, which result from reentrant corners or changing boundary conditions affect the development of the microstructure essentially. In both cases a phase consisting of the softer material develops in a vicinity of a point with a stress singularity.
Open Access
Modeling, analysis and simulation of 2D dynamic crack propagation
(2009) Lalegname, Adriana E.; Sändig, Anna-Margarete (Apl. Prof. Dr.)
The goal is to investigate mathematically the behavior of a linear elastic, isotropic, homogeneous and finite body with a running crack under the influence of a wave. Here we concentrate on bounded crack velocities. Reducing the 3D wave model given by Navier-Lame equation system to a 2D one we get an in-plane model for plane elastic waves and an out-of-plane model for shear waves. The main points are in both cases: the description of the behavior of the elastic fields near the running crack tip and the derivation of the equations of motion of the crack tip. The method: Analysis of the transformed problem. A well-tried method in solid mechanics is to transform the current configuration (a non-cylindrical space-time domain). For this purpose it will be assumed there is a family of mapping which maps the reference configuration (cylindrical domain) into the current configuration. Additionally a further configuration, where the isotropic Laplacian appears, is considered. Roughly speaking, $h(t)$ describes the motion of the crack tip. Performing the above change of variables we get elastic wave equations with time-dependent coefficients and lower order terms in the reference domain. Using functional analysis arguments the solvability of the transformed initial boundary value problem is studied. Derivation of the equation of motion: Starting from the rate of the total energy in the current configuration we have to derive an equation of motion of crack tip, that means, a nonlinear ordinary differential equation for $h(t)$. In order to calculate the rate of the contributing energies we transform again the integrals into the reference configuration and use the results of the first item. Most of the studies on the dynamic crack propagation from the viewpoint of the fracture mechanics postulate the body is infinite in a thickness direction because the mathematical treatment is simple. However this postulate is not pertinent to most practical cases. We report here an analytical study on the dynamic crack propagation in a finite configuration. For both cases, the out of- and in-plane, the analytical solution is determined in the vicinity of a moving crack located in an isotropic medium. As specially, we are concerned with the variation of stress intensity factors and displacement fields near the crack tip. In order to construct the asymptotic of the singular solutions for in-plane fracture case, the matching procedure will be used once more. In contrast to before, the method is not directly applicable. For getting these fields we introduce auxiliary potentials related to the in-plane motion of the crack which are separable into Mode I (opening mode) and Mode II (in-plane sliding mode) crack problems. Once such asymptotic fields near the tip of a propagating crack are determined, other important quantities of relevance in dynamic fracture mechanics, such as dynamic stress intensity factors and dynamic energy release rate can be determinate.
Open Access
Regularity results for quasilinear elliptic systems of power-law growth in nonsmooth domains : boundary, transmission and crack problems
(2005) Knees, Dorothee; Sändig, Anna-Margarete (Apl. Prof. Dr.)
The thesis is devoted to the analysis of weak solutions of quasi-linear elliptic boundary transmission problems on nonsmooth domains. The focus lies on the following two classes of equations which are closely related: general systems of second order quasi-linear elliptic partial differential equations of p-structure with piecewise constant coefficients (e.g. the p-Laplace equation) and field equations for the displacement and stress fields of heterogeneous, physically nonlinear elastic bodies which obey a constitutive relation of power-law type (Ramberg/Osgood model). In our context a heterogeneous body is a structure which is composed of different homogeneous sub-structures which are bonded along interfaces. The whole structure as well as the interfaces, which separate sub-structures with different material properties, may have corners and edges. Physical experiments and numerical simulations show that very high stress concentrations can occur in the vicinity of re-entrant corners, cracks, edges and near those points where the material parameters are discontinuous. These stress concentrations have a strong influence on the strength and physical life of the body and may finally lead to the failure of the whole structure. Therefore, information on the stress fields are important. The goal of the thesis is to derive global regularity results for weak solutions of the underlying boundary value and transmission problems. An example by R.B. Kellogg shows that one cannot guarantee general higher regularity for weak solutions without any further restriction on the geometry or the distribution of the material parameters. Therefore, we require that the boundary transmission problems satisfy a so-called quasi-monotone covering condition. This new condition imposes restrictions on the geometry of the subdomains and on the growth properties of differential operators on neighboured subdomains. Assuming that the quasi-monotonicity condition holds we prove global regularity results for weak solutions of quasi-linear elliptic boundary transmission problems of p-structure and for the displacement and stress fields of coupled Ramberg/Osgood materials. The results cover constellations with an arbitrary number of subdomains, nonsmooth boundaries and interfaces. Furthermore, the growth properties of the differential operators may vary from subdomain to subdomain. The results are proved with a difference quotient technique where the quasi-monotone covering condition and the growth properties of the differential operators play an essential role. We present two applications of the regularity results. We derive a global regularity result for the stress fields of Hencky's elasto-plastic model on domains with Lipschitz boundaries. The proof is based on a theorem by R. Temam and A. Bensoussan/J. Frehse stating that stress fields of the Hencky model can be approximated by stress fields of the Ramberg/Osgood model. As second application we study Griffith's energetic fracture criterion for Ramberg/Osgood materials. Here we use the regularity results for a mathematically rigorous justification of formulas (Griffith formula, J-integral) which occur in the formulation of Griffith's criterion.