Papers by Jüri Engelbrecht
Springer Proceedings in Physics, 2011
The dispersive wave motion in solids with microstructure is considered in the one-dimensional set... more The dispersive wave motion in solids with microstructure is considered in the one-dimensional setting in order to understand better the mechanism of dispersion. It is shown that the variety of dispersive wave propagation models derived by homogenization, continualization, and generalization of continuum mechanics can be unified in the framework of dual internal variables theory.
International Journal of Mechanical Sciences, Oct 1, 2015
Elastic wave propagation through diffraction gratings is studied numerically in the plane strain ... more Elastic wave propagation through diffraction gratings is studied numerically in the plane strain setting. The interaction of the waves with periodically ordered elastic inclusions leads to a self-imaging Talbot effect for the wavelength equal or close to the grating size. The energy localization is observed at the vicinity of inclusions in the case of elastic gratings. Such a localization is absent in the case of rigid gratings.
International Journal of Solids and Structures, 2013
The wave motion in micromorphic microstructured solids is studied. The mathematical model is base... more The wave motion in micromorphic microstructured solids is studied. The mathematical model is based on ideas of Mindlin and governing equations are derived by making use of the Euler-Lagrange formalism. The same result is obtained by means of the internal variables approach. Actually such a model describes internal fields in microstructured solids under external loading and the interaction of these fields results in various physical effects. The emphasis of the paper is on dispersion analysis and wave profiles generated by initial or boundary conditions in a one-dimensional case.
Proceedings of the Estonian Academy of Sciences, 2013
On the basis of the Mindlin-type micromorphic theory for wave motion in microstructured solids th... more On the basis of the Mindlin-type micromorphic theory for wave motion in microstructured solids the 1D governing equations and corresponding dispersion relations are derived. The leading physical dimensionless parameters are established and their importance for describing dispersion effects is discussed. The general discussion reveals the role of both geometrical and physical dimensionless parameters in mechanics of microstructured materials.
Universality of Nonclassical Nonlinearity, 2006
Wave propagation in microstructured materials is directly affected by the existence of internal s... more Wave propagation in microstructured materials is directly affected by the existence of internal space scale(s) in the compound matter. In this case the classical continuum theory cannot be used. In this paper based on the Mindlin model, the balance laws for macro-and microstructure are formulated separately. Using the slaving principles relating macro-and microdisplacements, the governing equations are derived for a single-and 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 having a clear physical background (microinertia, wave speed in microstructure) related to the scale of the microstructure. Full, approximated (corresponding to hierarchical models), and simplified dispersion relations are derived and analysed to demonstrate the validity of the hierarchical governing equations. Linear theory is based on the quadratic free energy function, in nonlinear theory the cubic terms should also be taken into account. The corresponding governing equation includes nonlinearities in both macro-and microscale. Such consistent modelling opens up new possibilities to Nondestructive Testing (NDT) of material properties.
Lecture Notes in Applied and Computational Mechanics, 2009
Dispersive wave propagation in solids with microstructure is discussed in the small-strain approx... more Dispersive wave propagation in solids with microstructure is discussed in the small-strain approximation and in the one-dimensional setting. It is shown that the generalizations of wave equation based on continualizations of discrete systems as well as on homogenization methods can be recovered in the framework of the internal variable theory in the case of non-dissipative processes.
Acta Mechanica, 2011
The basic ideas for describing the dispersive wave motion in microstructured solids are discussed... more The basic ideas for describing the dispersive wave motion in microstructured solids are discussed in the one-dimensional setting because then the differences between various microstructure models are clearly visible. An overview of models demonstrates a variety of approaches, but the consistent structure of the theory is best considered from the unified viewpoint of internal variables. It is shown that the unification of microstructure models can be achieved using the concept of dual internal variables.
Nonlinear Waves in Nonlocal Media
Applied Mechanics Reviews, 1998
This review article gives a brief overview on nonlocal theories in solid mechanics from the viewp... more 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.
Philosophical Magazine, 2005
The dispersive effects due to the presence of microstructure in solids are studied. The basic mat... more The dispersive effects due to the presence of microstructure in solids are studied. The basic mathematical model is derived following Mindlin's 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.
Proceedings of the Estonian Academy of Sciences, 2009
The Mindlin-type model is used for describing the longitudinal deformation waves in microstructur... more The Mindlin-type model is used for describing the longitudinal deformation waves in microstructured solids. The evolution equation (one-wave equation) is derived for the hierarchical governing equation (two-wave equation) in the nonlinear case using the asymptotic (reductive perturbation) method. The evolution equation is integrated numerically under harmonic as well as localized initial conditions making use of the pseudospectral method. Analysis of the results demonstrates that the derived evolution equation is able to grasp essential effects of microinertia and elasticity of a microstructure. The influence of these effects can result in the emergence of asymmetric solitary waves.
Abstract. The Mindlin-type model is used for describing the longitudinal deformation waves in mic... more Abstract. The Mindlin-type model is used for describing the longitudinal deformation waves in microstructured solids. The evolution equation (one-wave equation) is derived for the hierarchical governing equation (two-wave equation) in the nonlinear case using the asymptotic (reductive perturbation) method. The evolution equation is integrated numerically under harmonic as well as localized initial conditions making use of the pseudospectral method. Analysis of the results demonstrates that the derived evolution equation is able to grasp essential effects of microinertia and elasticity of a microstructure. The influence of these effects can result in the emergence of asymmetric solitary waves. Key words: nonlinear wave motion, microstructure, hierarchy of waves, evolution equations. 1.
IUTAM Bookseries, 2010
A linear model of the microstructured continuum based on Mindlin theory is adopted which can be r... more A linear model of the microstructured continuum based on Mindlin theory is adopted which can be represented in the framework of the internal variable theory. Fully coupled systems of equations for macro-motion and microstructure evolution are represented in the form of conservation laws. A modification of wave propagation algorithm is used for numerical calculations. Results of direct numerical simulations of wave propagation in periodic medium are compared with similar results for the continuous media with the modelled microstructure. It is shown that the proper choice of material constants should be made to match the results obtained by both approaches. T.-T. Wu and C.-C. Ma (eds.), IUTAM Symposium on Recent Advances of Acoustic
Estonian Journal of Engineering, 2013
The basic concepts for modelling wave propagation in solids with microstructure are described. It... more The basic concepts for modelling wave propagation in solids with microstructure are described. It is shown that the Green method, based on postulating the potential energy function, has certain advantages compared with the widely used Cauchy method, which postulates directly the stress-strain relations. Simple examples demonstrate how the Green method together with internal variables permits to determine the microstress and the interactive force between the constituents of solids. The structure of governing equations and possible physical effects captured by such modelling are described. The microstress and interactive force lead to the dispersion of waves at the macrolevel.
We reflect highlights of studies into a variety of phenomena reflecting the complexity of underly... more We reflect highlights of studies into a variety of phenomena reflecting the complexity of underlying nonlinear processes in a selection of research disciplines in the Centre of Nonlinear Studies (CENS), presented in the International Conference on Complexity of Nonlinear Waves, 5-7 October 2009, Tallinn, Estonia. We emphasize the similarity of mathematical description of and potential synergy arising from complementary studies in general soliton science, wave propagation in microstructured and functionally graded materials, related inverse problems, issues of nondestructive testing, weak resonant interactions of water waves, wave transformation and run-up, soliton interactions in shallow water, and selected problems of passive scalar turbulence.
Proceedings of the Estonian Academy of Sciences, 2018
Biological systems are characterized by many interwoven processes over multiple scales where inte... more Biological systems are characterized by many interwoven processes over multiple scales where interactions between the various phenomena play a decisive role. These phenomena can be chemical, electrical, and/or mechanical, all embedded into a whole. This paper discusses the propagation of signals in nerve fibres, which exhibits clear signs of complexity: electrical signals (action potentials) are coupled to mechanical waves in the internal axoplasmic fluid and in the surrounding biomembrane. Here the underlying microstructure affects strongly the processes: the existence of ion currents changes the balance of ions within fibres, the opening of ion channels in the surrounding biomembrane is a crucial process, and the biomembrane itself has a microstructure composed of lipids. The whole process is governed by interactions, and the analysis of single processes has demonstrated the importance of nonlinearities. The main challenge is to build up a general model where the coupling of all related phenomena is taken into account. It is proposed that three processes-the propagation of an action potential and mechanical waves in the biomembrane and in the axoplasmatic fluid-be united into a general model with additional interaction forces for reflecting coupling. Such a model results in the emerging of a mutually interacting ensemble of waves. The preliminary numerical simulations cast light onto the possible validation of this general model reflecting the complexity of signal propagation in nerve fibres. The mathematically consistent modelling will allow not only the prediction of process characteristics but gives a possibility of understanding the role of governing factors in the whole complex process.
Proceedings of the Estonian Academy of Sciences, 2015
A short overview on the research of Nikolai Alumäe (1915-1992) in the field of the theory of shel... more A short overview on the research of Nikolai Alumäe (1915-1992) in the field of the theory of shells is presented. His brilliant analytical results explaining the stability and transient processes in shells have not lost their importance although obtained in the 1950s and 1960s.
Mathematics and Mechanics of Complex Systems
This paper describes the mathematical models derived for wave propagation in solids with internal... more This paper describes the mathematical models derived for wave propagation in solids with internal structure. The focus of the overview is on one-dimensional models which enlarge the classical wave equation by higher-order terms. The crucial parameter in models is the ratio of characteristic lengths of the excitation and the internal structure. Novel approaches based on the concept of internal variables permit one to take the thermodynamical conditions into account directly. Examples of generalisations include frequency-dependent multiscale models, nonlinear models and thermoelasticity. The substructural complexity within the framework of elasticity gives rise to dispersion of waves. Dispersion analysis shows that acoustic and optical branches of dispersion curves together describe properly wave phenomena in microstructured solids. In the case of nonlinear models, the governing equations are of the Boussinesq type. It is argued that such models of waves in solids with microstructure ...
Wave Motion, 2011
The emergence of soliton trains and interaction of solitons are analyzed by using a Boussinesqtyp... more The emergence of soliton trains and interaction of solitons are analyzed by using a Boussinesqtype equation which describes the propagation of bi-directional deformation waves in microstructured solids. The governing equation in the one-dimensional setting is based on the Mindlin model. This model includes scale parameters which show explicitly the influence of the microstructure in wave motion. As a result the governing equation has a hierarchical structure. The analysis is based on numerical simulation using the pseudospectral method. It is shown how the number of solitons in emerging trains depends on the initial excitation. The head-on collision of emerged solitons is not fully elastic due to radiation but the solitons preserve their identity after collision and the speed of solitons is retained while the radiation keeps a certain mean value. That is why we have kept through this paper the notion of solitons.
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Papers by Jüri Engelbrecht