Jüri Engelbrecht

Waves in microstructured materials and dispersion

The dispersive effects due to the presence of microstructure in solids are studied. The basic mathematical model is derived following Mindlins theory. In the one-dimensional case the governing equations of a linear system are presented. An approximation using the slaving principle indicates a hierarchy of waves. The corresponding dispersion relations are compared with each other. The choice between the models can be made on the basis of physical effects described by dispersion relations.

Intracellular diffusion of adenosine phosphates is locally restricted in cardiac muscle

Recent studies have revealed the structural and functional interactions between mitochondria, myofibrils and sarcoplasmic reticulum in cardiac cells. Direct channeling of adenosine phosphates between organelles identified in the experiments indicates that diffusion of adenosine phosphates is limited in cardiac cells due to very specific intracellular structural organization. However, the mode of diffusion restrictions and nature of the intracellular structures in creating the diffusion barriers is still unclear, and, therefore, a subject of active research. The aim of this work is to analyze the possible role of two principally different modes of restriction distribution for adenosine phosphates (a) the uniform diffusion restriction and (b) the localized diffusion limitation in the vicinity of mitochondria, by fitting the experimental data with the mathematical model. The reaction-diffusion model of compartmentalized energy transfer was used to analyze the data obtained from the experiments with the skinned muscle fibers, which described the following processes: mitochondrial respiration rate dependency on exogenous ADP and ATP concentrations; inhibition of endogenous ADP-stimulated respiration by pyruvate kinase (PK) and phosphoenolpyruvate (PEP) system; kinetics of oxygen consumption stabilization after addition of 2 mM MgATP or MgADP; ATPase activity with inhibited mitochondrial respiration; and buildup of MgADP concentration in the medium after addition of MgATP. The analysis revealed that only the second mechanism considered – localization of diffusion restrictions – is able to account for the experimental data. In the case of uniform diffusion restrictions, the model solution was in agreement only with two measurements: the respiration rate as a function of ADP or ATP concentrations and inhibition of respiration by PK + PEP. It was concluded that intracellular diffusion restrictions for adenosine phosphates are not distributed uniformly, but rather are localized in certain compartments of the cardiac cells.

Numerical Simulation of Waves and Fronts in Inhomogeneous Solids

This book shows the advanced methods of numerical simulation of waves and fronts propagation in inhomogeneous solids and introduces related important ideas associated with the application of numerical methods for these problems. Great care has been taken throughout the book to seek a balance between the thermomechanical analysis and numerical techniques. It is suitable for advanced undergraduate and graduate courses in continuum mechanics and engineering. Necessary prerequisites for this text are basic continuum mechanics and thermodynamics. Some elementary knowledge of numerical methods for partial differential equations is also preferable.

Internal Variables and Dynamic Degrees of Freedom

Abstract Dynamic degrees of freedom and internal variables are treated in a uniform way. The unification is achieved by means of the introduction of a dual internal variable. This duality provides the corresponding evolution equations depending on whether the Onsager–Casimir reciprocity relations are satisfied or not.

Solitary waves in nonlinear microstructured materials

Dispersive effects due to microstructure of materials combined with nonlinearities give rise to solitary waves. In this paper the existence of solitary wave solutions is proved for a rather general hierarchical governing equation which accounts for nonlinearities on both macro- and microscales. Properties of the waves are established. Waves are asymmetric in the case of the nonlinearity in the microscale. Dispersive effects are due to the scale dependence.

On the KdV soliton formation and discrete spectral analysis

Abstract Based on the numerical integration (pseudospectral method) of the KdV equation governing the evolution of an initially monochromatic wave, a detailed picture of the soliton formation process is given. Systematic analysis of spectral properties of emerging KdV solitons over a wide range of dispersion parameters is presented. The number of emerging solitons is estimated from both the numerical experiment and the IST. The dynamics of spectral densities casts more light on the soliton formation process.

Hierarchies of Waves in Nonclassical Materials

Wave propagation in microstructured materials is directly affected by the existence of internal space scales in the compound matter. To allow for microstructures the classical continuum theory has to be generalized. In this chapter, the coupled balance laws for macroand microstructure based on the Mindlin model are formulated. Using the slaving principles relating macroand microdisplacements, the governing equations are derived for singleand two-scale (scale within scale) cases. These equations exhibit hierarchical properties assigning the wave operators to internal scales. In terms of macrodisplacements, higher-order dispersive terms appear that are related to the scale of the microstructure and reflect the effects of microinertia. The dispersion relations of propagating waves are established and compared with approximations resulting from hierarchical models and also with some simplified models. Linear theory is based on a quadratic free energy function, whereas in nonlinear theory cubic terms also are taken into account. The corresponding governing equation includes nonlinearities in both macroand microscale. This consistent modeling opens up new possibilities to NonDestructive Testing (NDT) of material properties.

Thermoelasticity with Dual Internal Variables

The extension of the thermoelasticity theory by weakly non-local dual internal variables enriched by an extra entropy flux for the thermomechanical description of the behavior of microstructured solids is presented. The internal variables take into account the distributed effect of microdeformations or microtemperatures (and their gradients) on the overall macroscopic behavior. The evolution equations for microtemperatures can be hyperbolic, which can induce wave-like propagation for macrotemperature due to the coupling of equations.

On the long-time behaviour of soliton ensembles

The paper is focused on the details of the emergence of Korteweg-de Vries (KdV) solitons from an initial harmonic excitation. Although the problem is a classical one, numerical simulations over a large range of dispersion parameters in the long run have demonstrated new features: the existence of soliton ensembles including also virtual (hidden) solitons and the periodic patterns of the wave-profile maxima.

Nonlinear Waves in Nonlocal Media

This review article gives a brief overview on nonlocal theories in solid mechanics from the viewpoint of wave motion. The influence of two essential qualities of solids—nonlocality and nonlinearity—is discussed. The effects of microstructure are analyzed in order to understand their role in nonlocal theories. The various models are specified on the level of one-dimensional unidirectional motion in order to achieve mathematical clarity of interpreting physical phenomena. Three main types of evolution equations are shown to govern deformation waves under such assumptions. Based on the dispersion analysis, weak, true, and strong nonlocalities are distinguished. There are 75 references included with this article.