Strain Solitary Waves Research Articles

Strain fields and solitary strain waves as determining factors for the cross-sectional geometry of mouse incisor enamel

Dental enamel in the mouse incisor is the subject of one of the most detailed histological records of cell motion and action during the formation and shaping of any organ in any species. We use the rich data to test the hypothesis that the shape of the enamel body on a perpendicular cross-section of the long, sabre-like incisor can be predicted by assuming that the formative ameloblast cells respond to strain and strain-rate cues that inform individual cells of position and time. The strain field is generated when growth of the forming enamel stretches the ameloblast population. Simultaneously, the strain is relaxed by coherent wavy cell movements. We hypothesize that wave motion arises when cells maintain homeostasis in their area density, with the rate of their recovery from a density perturbation assumed proportional to the magnitude of the perturbation. Density homeostasis gives rise to a nonlinear wave equation, which results in solitary waves propagating within computed strain fields. We predict the final thickness of the enamel by assuming ameloblasts stop generating enamel after they experience a critical strain condition. The thickness profile vs position is correctly determined to within a constant factor, which is the unknown rate constant in the wave equation. When the rate constant is calibrated by the peak amplitude of the thickness profile, the commencement of enamel formation (the onset of ameloblast secretion) vs position is then also correctly predicted by the passage of solitary waves, implying that the strain jump within the solitary wave may be the trigger for the onset of secretion.

Nonlinear wave localization in an acoustic metamaterial with attached masses through one element of main chain

The nonlinear strain solitary waves in a metamaterial mass-in-mass lattice model with a di-atomic symmetry is studied when attached masses tare located through one element of main chain are studied in the continuum limit. The reasonable variables are introduced to account for strains. An asymptotic procedure is developed to reduce coupled continuum equations to a single equation for longitudinal strains. A correspondence with the classic model with a di-atomic symmetry is established. An improvement of the model is suggested to account for a switch-on/off of the parts of the metamaterial model arising due to the studied kind of symmetry. It allows us to change the behavior of the localized nonlinear strain waves. This improvement is important for description of a durability of the metamaterial. Also it helps to understand new heat processes, like thermal diode, in complex symmetric lattices.

Nonlinear wave localization in an acoustic metamaterial

The nonlinear strain solitary waves in a metamaterial mass-in-mass lattice model are studied in the continuum limit. The main feature of the model is in the presence of the masses attached to the masses of the main chain giving rise to the coupling between the movement of the masses. An asymptotic procedure is developed to decouple the governing continuum equations and obtain the single equation for longitudinal strains. Analytical and numerical solutions of the equation reveal an influence of the nonlinearity of the metamaterial model and the attached mass on the type of strain waves localization and the number of generated localized waves. This, in turn, suggests an improvement of the model by an inclusion of a switch-on/off of the attached masses to provide a control of a nonlinear strain wave localization in the metamaterial.

Shock wave evolution into strain solitary wave in nonlinearly elastic solid bar

In this paper we present thorough experimental observation of the process of shock wave transformation into a bulk strain solitary wave in a nonlinearly elastic solid bar made of polystyrene. A theoretical model based on the describing propagation of a plane elastic wave in a bar is developed with account made for material nonlinearity and viscosity. Numerical modeling performed on the base of the developed model with and without regard to viscosity demonstrated formation of a long stable disturbance at proper account made for viscoelastic properties of the bar material.

AN EVALUATION OF SOLITARY WAVES FOR COMPACTION CONTROL ON FINE-GRAINED SOILS

This research presents the implementation of a non-destructive procedure for compaction control using a device capable of propagating solitary strain waves. The device is made of a chain of steel spheres excited by a strike at the top. The strain solitary wave is generated by a falling sphere with a constant height. The wave travels through the chain and is reflected when it reaches the boundary of the soil sample. The time gap between the incident and the reflected wave is called “time of flight”, and it is measured in the middle of the chain using a piezometric sensor. This “time of flight” isrelated to the mechanical properties of the tested specimen. The mechanical properties of soil are related to the density and water content that are achieved at the end of a compaction process. The mechanical properties can be used as an indicator for compaction control rather than density. The results suggest that the time of flight is an acceptable indicator to control the bulk density and optimum water content of compacted soils.

Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube

We first characterize strain solitary waves propagating in a fluid-filled membrane tube when the fluid is stationary prior to wave propagation and the tube is also subjected to a finite stretch. We consider the parameter regime where all traveling waves admitted by the linearized governing equations have nonzero speed. Solitary waves are viewed as waves of finite amplitude that bifurcate from the quiescent state of the system with the wave speed playing the role of the bifurcation parameter. Evolution of the bifurcation diagram with respect to the pre-stretch is clarified. We then study the stability of solitary waves for a representative case that is likely of most interest in applications, the case in which solitary waves exist with speed c lying in the interval [0, c_1) where c_1 is the bifurcation value of c, and the wave amplitude is a decreasing function of speed. It is shown that there exists an intermediate value c_0 in the above interval such that solitary waves are spectrally stable if their speed is greater than c_0 and unstable otherwise.

On Boussinesq-type models for long longitudinal waves in elastic rods

In this paper we revisit the derivations of model equations describing long nonlinear longitudinal bulk strain waves in elastic rods within the scope of the Murnaghan model in order to derive a Boussinesq-type model, and extend these derivations to include axially symmetric loading on the lateral boundary surface, and longitudinal pre-stretch. We systematically derive two forced Boussinesq-type models from the full equations of motion and non-zero surface boundary conditions, utilising the presence of two small parameters characterising the smallness of the wave amplitude and the long wavelength compared to the radius of the waveguide. We compare the basic dynamical properties of both models (linear dispersion curves and solitary wave solutions). We also briefly describe the laboratory experiments on generation of bulk strain solitary waves in the Ioffe Institute, and suggest that this generation process can be modelled using the derived equations.

Scattering of bulk strain solitary waves in bi-layers with delamination

We study the scattering of longitudinal bulk strain solitary waves in delaminated bi-layers with different types of bonding. The direct numerical modelling of these problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and the Integrability Theory of the Korteweg - de Vries equation by the Inverse Scattering Transform. We show that the semi-analytical approach agrees well with the direct numerical simulations and use it to study the scattering of different types of longitudinal bulk strain solitary waves in a wide range of bi-layers with delamination. In particular, we model the dynamics of a long longitudinal strain solitary wave in a symmetric perfectly bonded bi-layer with delamination. The numerical modelling confirms that delamination causes fission of an incident solitary wave and, thus, can be used to detect the defect. We then extend our approaches to the modelling of the waves in bi-layers with soft (“imperfect”) bonding, described by a system of coupled Boussinesq equations and supporting radiating solitary waves. The results may help us to control the integrity of layered structures.

On radiating solitary waves in bi-layers with delamination and coupled Ostrovsky equations.

We study the scattering of a long longitudinal radiating bulk strain solitary wave in the delaminated area of a two-layered elastic structure with soft ("imperfect") bonding between the layers within the scope of the coupled Boussinesq equations. The direct numerical modelling of this and similar problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and averaging with respect to the fast space variable, leading to the coupled Ostrovsky equations in bonded regions and uncoupled Korteweg-de Vries equations in the delaminated region. We show that the semi-analytical approach agrees well with direct numerical simulations and use it to study the nonlinear dynamics and scattering of the radiating solitary wave in a wide range of bi-layers with delamination. The results indicate that radiating solitary waves could help us to control the integrity of layered structures with imperfect interfaces.

Modelling of nonlinear wave scattering in a delaminated elastic bar.

Integrity of layered structures, extensively used in modern industry, strongly depends on the quality of their interfaces; poor adhesion or delamination can lead to a failure of the structure. Can nonlinear waves help us to control the quality of layered structures? In this paper, we numerically model the dynamics of a long longitudinal strain solitary wave in a split, symmetric layered bar. The recently developed analytical approach, based on matching two asymptotic multiple-scales expansions and the integrability theory of the Korteweg–de Vries equation by the inverse scattering transform, is used to develop an effective semi-analytical numerical approach for these types of problems. We also employ a direct finite-difference method and compare the numerical results with each other, and with the analytical predictions. The numerical modelling confirms that delamination causes fission of an incident solitary wave and, thus, can be used to detect the defect.

Nonlinear elastic waves in layered bars made of different materials

The paper presents experimental observation of strain solitary waves in layered waveguides with layers made of two glassy polymers, PMMA and polystyrene. It is shown that in delaminated structures solitons propagate in each layer independently with parameters typical for waves in each material. In bars with layers bonded with the glassy adhesive a combined wave is formed, propagating as a single one in both layers and exhibiting soliton properties.

Strain solitary waves in a thin-walled waveguide

A mathematical model is proposed to describe bulk longitudinal waves in a nonlinearly elastic thin-walled cylindrical shell. The equation of motion for the longitudinal displacement is derived. In case of the homogeneous shell, this equation is reduced to the doubly dispersive equation for the linear longitudinal strain component and provides a solitary wave solution. Results of the first experimental observation of the bulk strain soliton in a duct-like shell are presented, and both the wave amplitude and velocity are estimated.

Bulk strain solitary waves in bonded layered polymeric bars with delamination

We report the registration of delamination induced variations in the dynamics of bulk strain solitary waves in layered polymeric bars with the glassy and rubber-like adhesives, for the layers made of the same material. The key phenomenon in a layered structure with the glassy bonding is the delamination caused fission of a single incident soliton into a wave train of solitons, with the detectable increase in the amplitude of the leading solitary wave. The significant feature of bulk strain solitons in structures bonded with the rubber-like adhesive is the generation of radiating solitary waves, whilst co-propagating ripples disappear in the delaminated area. The observed variations may be used for the detection of delamination in lengthy layered structures.

Longitudinal Strain Solitary Wave in a Two‐Layered Polymeric Bar

Abstract: Both theoretical investigations and successful experimental research were performed recently, confirming the existence and demonstrating the main properties of bulk strain solitary waves in nonlinearly elastic solid wave guides. Our current research is devoted to nonlinear wave processes in layered elastic wave guides with inhomogeneities modelling damage/delamination. Here, we present first experimental and theoretical results, demonstrating the change in the amplitude and width of a strain solitary wave propagating in an inhomogeneous two‐layered wave guide made of Polymethyl Methacrylate (PMMA). Parameters of such waves in layered structures may be useful for the assessment of their operational integrity and robustness.

Study on Strain Solitary Waves in the Nonlinear Elastic Circular Rod

Taking into account the nonlinear constitutive relationship and transverse Poisson effects, the propagation characteristic of nonlinear wave for one-dimension elastic thin rod is studied. With the help of Mathematic, two traveling wave solutions for this nonlinear wave equation are obtained by sine-cosine function method, which include the shock wave solution and the solitary wave solution. The necessary condition of these solutions is given also.

Formation of strain solitary waves in the Cosserat continuum with restricted rotation

Formation of strain solitary waves in the Cosserat continuum with restricted rotation

Comparison of the effect of cyanoacrylate- and polyurethane-based adhesives on a longitudinal strain solitary wave in layered polymethylmethacrylate waveguides

We study the effect of two types of adhesive bonding on the propagation of a localized longitudinal strain wave in two- and three-layered elastic bars. The detectable variation in the decay rate of the wave at relatively long distances of propagation is observed. It is proposed that such variation can be used as an indicator of the type of an interface.

Do the bulk solitary strain waves decay?

Non-linearly elastic materials may provide generation of strain solitons, even under short-run and reversible (elastic) loading. Theoretical and experimental studies have been performed to prove the existence of long bulk strain solitary waves produced by a laser-induced shock wave in non-linearly elastic isotropic wave guides. Propagation of the bulk solitary density waves is studied in lengthy (over 0.5 m long) elastic wave guides, which have strong dispersion and are made of polymers with remarkable decay of any of the linear or shock waves at short distances. New experiments in lengthy bars confirm that the bulk solitons do not reveal any considerable decay and shape transformation, while linear or shock waves disappear at much shorter distance completely. Decrement values are estimated first for elastic non-linear strain waves decay in polymeric bars.

Fission of a longitudinal strain solitary wave in a delaminated bar

The aim of the paper is to show that splitting of a waveguide leads to fission of bulk solitons in solids. We study the dynamics of a longitudinal bulk solitary wave in a delaminated, symmetric layered elastic bar. First, we consider a two-layered bar and assume that there is a perfect interface when x<0 and complete debonding (splitting) when x>0 , where the axis Ox is directed along the bar. We derive the so-called doubly dispersive equation (DDE) for a long nonlinear longitudinal bulk wave propagating in an elastic bar of rectangular cross section. We formulate the problem for a delaminated two-layered bar in terms of the DDE with piecewise constant coefficients, subject to continuity of longitudinal displacement and normal stress across the "jump" at x=0 . We find the weakly nonlinear solution to the problem and consider the case of an incident solitary wave. The solution describes both the reflected and transmitted waves in the far field, as well as the diffraction in the near field (in the vicinity of the jump). We generalize the solution to the case of a symmetric n -layered bar. We show that delamination can lead to the fission of an incident solitary wave, and obtain explicit formulas for the number, amplitudes, velocities, and positions of the secondary solitary waves propagating in each layer of the split waveguide. We establish that generally there is a higher-order reflected wave even when the leading order reflected wave is absent.

Strain solitary wave propagation in laser‐excited solids

AbstractNonlinear dynamics of long 1D longitudinal waves in laser‐excited plates is investigated by taking into account the interaction of the longitudinal displacements with the temperature field and the field of concentration of non‐equilibrium (recombining) atomic point defects. The governing nonlinear equation is derived for describing the self‐consistent thermoelastic strain fields. It is found that the thermoelastic effect on the strain waves manifests itself in the appearance of dissipative terms, which describe the heat transfer and the thermoe lastic interaction caused by the strain‐induced heat releasedue to the recombination of atomic defects. The equation that describes the evolution of the amplitude of nonlinear localized waves with time in the single‐wave approximation is derived, and, on the basis of this equation, the damping increments of these waves are determined with allowance the dissipative losses. The effect of interaction of counterpropagating waves is considered. The influence of stress‐induced decay of defect complexes on the evolution of nonlinear dispersive strain waves is analyzed. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)