Microstructured Solids Research Articles

On the propagation of 1D solitary waves in Mindlin-type microstructured solids

The Mindlin model and hierarchical approach by Engelbrecht and Pastrone are used for modelling 1D wave propagation in microstructured solids. After introducing the free energy function, one gets from Euler–Lagrange equations a system of equations of motion. Making use of the slaving principle, a nonlinear hierarchical wave equation can be derived. Equations are solved numerically under localized initial conditions. For numerical integration, the pseudospectral method based on the Fourier transform is used. The influence of free energy parameters on the character of dispersion and wave propagation is studied. Numerical results of hierarchical approximation and the full equation system will be compared and the quality of the approximation will be discussed.

Nonlinear wave motion and complexity

Contemporary complexity science deals with problems which involve many variables interacting with each other in such a way that a new quality appears. An important cornestone of complex systems is nonlinearity. In this paper nonlinear wave motion in microstructured solids is analysed from the viewpoint of complexity. The basic models are derived by using the concept of internal variables which are related to dissipation inequality. The scale dependence results in wave hierarchies where dispersive effects are important. In such nonlinear models solitary wave structures may emerge – a typical sign of a new quality characteristic of complexity. In addition, examples from biophysics are presented, which demonstrate clearly the similarity to the ideas of complexity shown for waves in solids.

On periodic waves governed by the extended Korteweg–de Vries equation

Janno and Engelbrecht [1] have shown that the propagation of nonlinear one-dimensional waves in microstructured solids leads to a generalized Korteweg-deVries equation. After suitable transformation of the variables, the evolution equation of the waves can be written in the form [2] yt + y 2 � x + yxxx + 3 y 2 � xx = 0. (1) The additional term involving the small parameter reflects the nonlinearity in the microscale. Solutions representing asymmetric solitary waves have been analyzed both qualitatively and numerically in [1]. An approximate solution in analytic form has been given in [2]. Solitary waves can be considered as the long-wave limit of periodic solutions which, in the KdV case, have the form of cnoidal waves. The present paper is devoted to periodic solutions y = q(x ct) of the extended KdV equation (1), which emerge from the cnoidal waves for > 0.

On the propagation of solitary waves in Mindlin-type microstructured solids

On the propagation of solitary waves in Mindlin-type microstructured solids

On solitary waves in one‐dimensional microstructured solids

AbstractIn a microstructured solid with nonlinearities in macroscale and microscale, the propagation of 1D waves is governed by an extended KdV equation, which admits asymmetric solitary waves. An approximate solution is presented. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On one-dimensional solitary waves in microstructured solids

For describing the longitudinal deformation waves in microstructured solids a Mindlin-type model is used. The embedding of a microstructure in an elastic material is reflected in an inherent length scale causing dispersion of propagating waves. Nonlinear effects, if taken into account, will counteract dispersion. A suitable balance between nonlinearity and dispersion may permit the propagation of solitary waves. Following previous work by Engelbrecht and others the nonlinear hierarchical model is derived in a one-dimensional setting which corresponds to a two-wave equation. The evolution equation as a simplified model, representing a one-wave equation, is able to grasp the essential effects of microinertia and elasticity of the microstructure. It is shown that the nonlinearity in microscale leads to an asymmetry of the wave profile. The nonlinear evolution equation as an extended Korteweg–de Vries equation is solved approximately by a series expansion in a small parameter representing the micro-nonlinearity. Already the first approximation indicates the asymmetry of the solitary waves. It is shown that solitary waves will propagate only if the micro-nonlinearity does not exceed some upper bound. For the limiting case, an analytical solution of the extended Korteweg–de Vries equation can be provided and used as a reference for the approximate solutions.

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin–Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein–Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.

IDENTIFICATION OF MICROSTRUCTURED MATERIALS BY PHASE AND GROUP VELOCITIES

An inverse problem to determine parameters of microstructured solids by means of group and phase velocities of wave packets is studied. It is proved that in the case of normal dispersion the physical solution is unique and in the case of anomalous dispersion two physical solutions occur. Numerical tests are presented.

On modelling wave motion in microstructured solids

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.

Mixed finite element formulation for the general anti-plane shear problem, including mode III crack computations, in the framework of dipolar linear gradient elasticity

A mixed formulation is developed and numerically validated for the general 2D anti-plane shear problem in micro-structured solids governed by dipolar strain gradient elasticity. The current mixed formulation employs the form II statement of the gradient elasticity theory and uses the double stress components and the displacement field as main variables. High order, C0-continuous, conforming basis functions are employed in the finite element approximations (p-version). The results for the mode III crack problem reveal that, with proper mesh refinement at the areas of high solution gradients, the current approximation method captures the exact solution behaviour at different length scales, which depend on the size of material micro-structure. The latter is of vital importance because, near the crack tip, the nature of the exact solution, changes radically as we proceed from the macro- to micro-scale.

Inverse problems related to a coupled system of microstructure

Inverse problems to determine parameters of microstructured solids are studied. These parameters enter as coefficients to a coupled system of 1D wave propagation derived from the Mindlin model. Characteristics of harmonic waves and wave packets are used as data for the inverse problems. Uniqueness of the solution to the inverse problems in the case of normal dispersion is shown. In the case of anomalous dispersion the solution is double.

Numerical simulation of interaction of solitary deformation waves in microstructured solids

In the present paper 1D wave propagation in microstructured solids is modelled based on the Mindlin theory and hierarchical approach. The governing equation under consideration is non-integrable therefore it is analysed numerically. Propagation and interaction of localised initial pulses is simulated numerically over long time intervals by employing the pseudospectral method. Special attention is paid to the solitonic character of the solution.

On modelling dispersion in microstructured solids

The Mindlin-type model is used for describing the longitudinal deformation waves in microstructured solids. A simplified hierarchical model is derived in one-dimensional setting which is a two-wave equation. In addition, the evolution equations (one-wave equations) are derived for both the full and simplified models. It is shown that the simplified model as well as evolution equations grasp main effects of dispersion in a wide range of physical parameters.

On the interaction of deformation waves in microstructured solids; 93–99

On the interaction of deformation waves in microstructured solids; 93–99

On the propagation of solitary pulses in microstructured materials

Abstract KdV-type evolution equation, including the third- and the fifth-order dispersive and the fourth-order nonlinear terms, is used for modelling the wave propagation in microstructured solids like martensitic–austenitic alloys. The character of the dispersion depends on the signs of the third- and the fifth-order dispersion parameters. In the present paper the model equation is solved numerically under localised initial conditions in the case of mixed dispersion, i.e., the character of dispersion is normal for some wavenumbers and anomalous for others. Two types of solution are defined and discussed. Relatively small solitary waves result in irregular solution. However, if the amplitude exceeds a certain threshold a solution having regular time–space behaviour emerges. The latter has tree sub-types: “plaited” solitons, two solitary waves and single solitary wave. Depending on the value of the amplitude of the initial pulse these sub-types can appear alone or in a certain sequence.

Waves in microstructured solids: Inverse problems

Microstructured solids are characterized by their dispersive properties. Here inverse problems for such dispersive solids are analyzed based on the one-dimensional wave propagation. Governing equation of the Mindlin type model is considered which consists of higher order derivatives responsible for dispersive effects. Two cases are analyzed. The first deals with harmonic waves, the second – with a localized harmonic excitation. The dispersion relations are derived and then the corresponding inverse problems stated and solved. It is shown what information can be extracted from known (measured) phase and group velocities for determining the physical parameters. The results can be used in the nondestructive testing (NDT).

On solitons in microstructured solids and granular materials

The problems under consideration are related to wave propagation in nonlinear dispersive media, characterised by higher-order nonlinear and higher-order dispersive effects. Particularly two problems — wave propagation in dilatant granular materials and wave propagation in shape-memory alloys — are studied. Model equations are KdV-like evolution equations in both cases. The types of solutions are determined and analysed. It has been found that waves in granular materials are composed by two concurrent ensembles of solitary waves and there exist a threshold parameter value governing the behaviour of stable waves in alloy type materials.

Wave Propagation in Microstructured Solids

A general model of bodies with vectorial microstructures is introduced, which includes many particular models such as scalar and affine microstructures, Cosserat continua, and liquid crystals. Assuming the existence of a strain energy function, the field equations are obtained via a variational principle as Euler-Lagrange equations of an energetic functional. Dissipative forces are also introduced. Equilibrium and stability problems are briefly discussed. Within the framework of singular surfaces theory, nonlinear acceleration and shock waves are studied and proper propagation conditions for nondissipative materials are given. The influence of dissipation is briefly discussed.

Propagation of sech2-type solitary waves in higher-order KdV-type systems

Abstract Wave propagation in microstructured media is essentially influenced by nonlinear and dispersive effects. The simplest model governing these effects results in the Korteweg–de Vries (KdV) equation. In the present paper a KdV-type evolution equation, including the third- and fifth-order dispersive and the fourth-order nonlinear terms, is used for modelling the wave propagation in microstructured solids like martensitic–austenitic alloys. The model equation is solved numerically under localised initial conditions. Possible solution types are defined and discussed. The existence of a threshold is established. Below the threshold, the relatively small solitary waves decay in time. However, if the amplitude exceeds a certain threshold, i.e., the critical value, then such a solitary wave can propagate with nearly a constant speed and amplitude and consequently conserve the energy.

Mutually induced variations in dissipation and elasticity for oscillations in hysteretic materials: Non-simplex interaction regimes

Self-action and effects mutually induced by oscillations interacting in hysteretic media are investigated analytically and numerically. Special attention is paid to non-simplex processes for which presence of intermediate extrema results in appearance of minor nested loops inside the main hysteretic stress–strain loop. Non-simplex regimes are typical of interaction of excitations having different frequencies and amplitudes, but comparable strain rates. It is found that, due to transition between the regimes, frequency and amplitude dependencies of the variations in elasticity and dissipation induced by one wave for another one may become non-monotonous. Either additional dissipation or induced transparency may occur in different regimes. The results obtained are important for correct interpretation of experimental data on nonlinear acoustic interactions in rocks and many other microstructured (mesoscopic) solids that are known to exhibit elastic hysteresis and memory properties.