Applications of Homogenization Theory to the Study of Mineralized Tissue functions as an introduction to the theory of homogenization. At the same time, the book explains how to apply the theory to various application problems in biology, physics and engineering.
This useful research monograph is suitable for applied mathematicians, engineers and geophysicists. As for students and instructors, it is a well-rounded and comprehensive text on the topic of homogenization for graduate level courses or special mathematics classes.
Features:
Theory of homogenization
The main topic of the research work is concerned with the theory of homogenization and its application to statistically uniform materials, a group of materials for which a so-called representative volume element (RVE) can be defined. The approach is based on the idea of defining micro- and macro boundary value problems (BVP) which are related to each other by using the principle of the volume average and the Hill-Mandel macrohomogeneity condition. The latter requires the equality of the macrowork with the volume average of the microwork and is used to define the boundary conditions for the RVE.
Inverse FE^{2 }analysis
In many cases, the microstructure of composite materials is not known and cannot directly be accessed such that an inverse analysis is necessary for its investigation. This approach requires the implementation of two tools: an optimization method for the minimization of the error problem and a mechanical approach for the solution of the direct problem, i.e. the simulation of composite materials. One particular choice deals with the combination of the Levenberg–Marquardt method with the multiscale finite element method. The typical examples in this field investigate the elastic parameters for multi-phase materials. The sensitivity with respect to the initial guess and the influence of the measurement error are common problems to determine a unique solution.
Current trends towards lightweight design and the production of individualized and functionalized components often lead to multi-material products that combine several materials. For some material combinations, fusion welding processes are not applicable due to the fact that the materials to be joined have largely different melting points or will show metallurgical reactions that are detrimental for the product properties. Metallic multi-material products can be joined by plastic deformation in these cases. Examples of joining-by-forming processes are roll-bonding and friction-welding. In such processes, interdiffusion, evolution of the interface properties and microstructure evolution of both contacting solid bodies by plastic deformation, recovery, recrystallization, grain growth and phase transformations may take place concurrently and with strong interaction of the individual processes. The roll bonding of copper-aluminum composites deserves special attention because of the radically different behavior of the component materials: whereas DRX is typical of Cu, the dynamic recovery occurs in Al. Such Cu-Al-claddings are used in conductor wires and joints for conductor wires. Replacing copper partly by aluminum offers various advantages such as high conductivity at a lower weight and cost compared to monolithic copper.
Modeling of polymers
This working field deals with a continuum mechanical model for the curing of polymers, including the incompressibility effects arising at the late stages of the process. For this purpose, the free energy density functional is split into a deviatoric and a volumetric part, and a multifield formulation is inserted. An integral formulation of the functional is used to depict the time-dependent material behavior. In addition, the attention is paid to the modeling of viscoelastic and shrinkage effects. For the modeling of viscous effects, the deformation at the microlevel is decomposed into an elastic and a viscoelastic part, and a corresponding energy density consisting of equilibrium and non-equilibrium parts is proposed. In contrast to the viscous effects, the modeling of shrinkage effects does not require any further extension of the expression for the energy density, but an additional decomposition of the deformation into a shrinkage and a mechanical part. The model is also coupled with the multiscale finite element method in order to simulate the behavior of reinforced polymers.
Modeling of diffusion processes at the boundary of crystals
The solution-precipitation creep is believed to be the leading deformation process in the subduction zone, and thus responsible for plate tectonics. The process shows similarities to the Coble creep with the difference that the material transport takes place in intercrystalline space and not along the boundary of crystals. For its modeling, a variational approach has been devised in which the elastic energy remains in the standard form but a novel, specific formulation of the dissipation functional is proposed.
Modeling of wave propagation through fluids and elastic solids
In this research field, the focus is on the harmonic excitation and modeling of viscous effects. For this purpose, a formulation in the complex domain is assumed. This approach is illustrated on the basis of the modeling of the cancellous bone and the investigation of the process of osteoporosis. The cancellous bone is a specific tissue consisting of the solid skeleton and the fluid marrow for whose laboratory investigation ultrasonic procedures are typically used. These experiments are also subjects of simulations, particularly applied to calculate the attenuation coefficient which depends on the bone density.
Software - Multiscale FE program MSFEAP
As the main result of the work on the above topics, the multiscale FE program MSFEAP has been written. This program uses the FE program FEAPpv ( Robert L. Taylor , University of California, Berkeley) as a basis. Its extension, the MSFEAP program, is suitable for simulating heterogeneous materials. The user interface and commands specific to the original program remain unchanged with the difference that the commands can be applied at two levels. A further extension of the program by implementing new elements is easily possible due to its modular structure. In order to clarify the basics of the homogenization theory and the application of the program, a user manual including characteristic examples is compiled. The complete input files for the described problems are also provided.
Current Projects
DFG Project - Multiscale modeling of calcified polymer hydrogels(Start: July 2020)PI: S. KlingeCoworker: S. Aygün
Hydrogels, a significant group of highly hydrated polymers, represent the best choice for the potential application to bone fracture regeneration, which goes back to their bioactivity, affinity for biologically active proteins and compatibility with the bone tissue. However, this kind of materials also shows a serious disadvantage, namely, it loses its mechanical strength through swelling. This makes its straightforward usage difficult and motivates the development of different enhancement procedures. One of the most modern techniques for this purpose is calcification or, in a more general sense, mineralization. This method is inspired by the natural process of the bone growth where the enzyme alkaline phosphatase causes mineralization of the bone by cleavage of the phosphate from organic molecules. An analogous process induces homogeneous mineralization of a hydrogel and increases its mechanical strength. Recently, optical and electron microscopy has revealed that calcification yields different types of microstructure dependent on the type of the underlying polymer, and thus has clearly indicated that computational modeling can significantly contribute to the targeted investigation of effective behavior and material parameters. Fracture energy and diffusivity are two particularly important aspects in this context. The former is taken as the main measure of material ductility and represents a weak point of calcified hydrogels. In order to solve this challenging problem, inspiration once more comes from natural materials and their hierarchical microstructure. The study of diffusion in macromolecular solutions is motivated by many biomedical applications as well as by its key role for protein assembly and interstitial transport. The project furthermore studies the design of the mineralization process which includes two essential steps: the understanding of the mechanisms governing the microstructure development and subsequently their optimization. The investigation of the diffusivity and of mineralization requires a profound knowledge on the processes on the nanoscale. This of course strongly substantiates computer simulations, since this kind of processes is yet non-accessible even by the most modern microscopy techniques. The spectrum of applicable methods encompasses the multiscale finite element method, the phase field method, the model reduction strategy and the finite difference method.
D-A-CH Project (DFG, FWF) - Computational Modeling of Vesicle-Mediated Cell Transport (CM-TransCell)(Start: March 2018)PIs: S. Klinge and G. A. HolzapfelCoworkers: T. Wiegold and D. Haspinger
The particularly important characteristics of eukaryotic cells are the enormous complexity of their membrane anatomy and the high level of organization of the transport processes. The surprisingly precise manner of the routing of vesicles to various intracellular and extracellular destinations can be illustrated by numerous examples such as the release of neurotransmitters into the presynaptic region of a nerve cell and the export of insulin to the cell surface.
The key idea of the present project is to couple results of biomedical investigations and mechano-mathematical models with the highly efficient engineering software packages in order to simulate this type of processes, in particular the vesicle transport. The results should bridge the theoretical investigations and medical praxis and shift the paradigm in understanding and remedying different diseases, which certainly is the primary and long-term goal of the project. The individual objectives coincide with the modeling of single aspects of the vesicle transport, namely with the simulation of mechanisms by which the vesicles form, find their correct destination, fuse with organelles and deliver their cargo. The application of several different approaches is envisaged for this purpose, but three main strategies build the underlying skeleton: the theory of lipid bilayer membranes, the homogenization method and the diffusion theory. The mentioned approaches will furthermore be combined with the modern numerical techniques such as the finite element method and the multiscale finite element method.
In the final stage, the realization of single objectives will allow the simulation of vesicle transport as a continuous process and the study of the impact of various factors on the whole process. This way, the project will yield a significant shift from "static" bio-computations related to the single cell compartments and substeps of its activities, to the "dynamic" simulation of the real living processes.
Completed Projects
The objective of subproject C04 is the development of a micromechanical model for polycrystals which shall be able to consistently simulate plastic deformations together with damage. To this end, an extended crystal plasticity model able to simulate the damage within single crystals is proposed. It is furthermore combined with an interface model in order to additionally capture influences of damage on the grain boundaries. Finally, both material models are applied in order to simulate the behavior of an appropriately chosen RVE and to study the influence of initial damage.
DFG Project - Multiscale Modeling of Strain-Induced Crystallization in Polymers (MM-SIC)PI: S. KlingeCoworker: S. Aygün
The present project treats a polymer affected by the strain induced crystallization (SIC) as a heterogeneous medium consisting of regions with the different degree of network regularity. Such a concept allows depicting the nucleation and the growth of crystalline regions as well as the change of effective material parameters depending on the level of the strain applied. The model proposed is thermodynamically consistent. It is based on the assumptions for the free Helmholtz energy and dissipation. Both of them primarily include bulk- and surface terms due to the deformation and crystallization. The external variables are deformations and temperature, whereas the inelastic deformations and degree of the network regularity are internal variables. Their evolution equations are derived according to the principle of maximum of dissipation. The influences of latent heat and of temperature change are implemented in order to simulate thermal effects. The explained framework is advantageous for several reasons. First, it is suitable to answer the crucial question of which process predominantly influences SIC: the nucleation of new crystalline regions or the growth of already existing ones. Secondly, the proposed model is ideal for a direct implementation within the standard multiscale finite element concept. This numerical homogenization procedure is compatible with the theory of finite strains and is applicable for modeling the cases where the ratio of characteristic lengths of scales tends to zero. Both of these features are necessary for the effective modeling of SIC. The project also includes a study of stochastic aspects of the process, where a distribution function for the observable variables is introduced to express the expectation value of relevant quantities. The necessary evolution equation is derived by considering the effective energy of a control volume. The main goals here are to study nucleation and to evaluate the average size of the regions with different regularities of the network. The solution of the tasks itemized will make it possible to achieve the final project goal: the advanced simulations of SIC which can significantly contribute to the more efficient designing and usage of polymers. This is especially motivated by the fact that SIC has to be understood as a kind of reinforcement already successfully applied for some rubber materials. The proposed concepts are of general nature and can be taken as a basis for the modeling of similar processes involving the evolution of the internal microstructure.
List of publications
Publications in journals (reviewed)
Publications in journals
Contributions in books and in proceeding books