Semi-infinite Bar Research Articles

Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. This mixed problem models the propagation of displacement waves during a longitudinal impact on a bar, when the load remains in contact with the bar and the bar has a linear elastic element at the end. On the lower boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. The boundary condition, including the unknown function and its first and second order partial derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proven and the conditions are established under which a piecewise-smooth solution exists. The problem with matching conditions is considered.

Improved soil–structure interaction model considering time-lag effect

An important aspect of soil– structure interaction analysis is establishing a high-precision conversion relationship between the dynamic impedance in the frequency domain and impulse response function in the time domain. Accordingly, an improved Nakamura model was developed in this study to convert dynamic impedance to an impulse response function. The causes and distribution rules of repeated phase angles were investigated based on the traveling time-lag effect, rendering the improved method more robust. In addition, the impulse response function and interaction force can achieve high fitting accuracy by stripping the singular term from the conventional term. In this study, the time-domain impulse response function and interaction force were derived based on the force–displacement relationship. Further, the accuracy and stability of the improved method were verified using a semi-infinite bar on an elastic foundation. Then, the effects of frequency interval, maximum frequency, and shape of the coefficient matrix on the impulse response function and interaction force were studied by considering the analytical solution of a spherical cavity embedded in full space as an example. Finally, two interaction systems were used to verify the applicability of the improved method in analyzing the soil–structure dynamic interaction.

Wave propagation in a non-linearly elastic bar with phase transformations

There exist considerable difficulties in studying the wave propagation problems of the kind of phase transforming materials with non-monotone material responses due to the lack of uniqueness of solution for a moving phase boundary. An admissibility condition, which is named the Maxwell equal-area criterion, was established from a three-dimensional (3D) internal-variable formulation in the previous study to supplement the one-dimensional (1D) dynamical system of the bars/rods made of the special materials to determine the phase boundary uniquely. In this paper, we employ this criterion to investigate the wave propagation in the bar made of the special kind of phase transforming materials, which are associated with trilinear non-monotone stress–strain responses. The Riemann problem for different boundary states that arises from wave encounters are analyzed by the method of wave curves. Based on the construction of the Riemann solution, we explicitly study the wave propagation in a semi-infinite bar under an impulsive stress impact to explore the dynamical features of the special materials. The wave structures in the entire spatial and temporal domain for two different impact levels are obtained, as well as the state profiles. Detailed analyses and discussions of the wave patterns and wave encounters occurring during the wave propagation process are given for a deep understanding of dynamical phase transitions of the materials. These results are further compared with those obtained using the artificially adopted maximal dissipation rate criterion. The similarities and differences are discussed. The comparisons further confirmed that the special phase transforming material can be used for designing wave-absorption/impact-protection devices.

The classical solution of one problem of an absolutely inelastic impact on a long elastic semi-infinite bar

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom boundary, the Cauchy conditions are specified, meanwhile, the second of them has a discontinuity of the first kind at one point. The smooth boundary condition, which has the first and the second order derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved and the conditions are established under which a piecewise-smooth solution exists. The problem with matcing conditions is considered.

Three-dimensional hybrid asynchronous perfectly matched layer for wave propagation in heterogeneous semi-infinite media

This paper presents an efficient hybrid asynchronous three-dimensional (3D) perfectly matched layer (PML) for modeling unbounded domains. The proposed unsplit explicit or implicit 3D PML formulation is implemented in the framework of a heterogeneous asynchronous time integrator. It is fully versatile in terms of time integrators and time step sizes according to partitions while conserving classical finite element formulations in the elastic domain without complex-valued stretched coordinates. Examples of a semi-infinite bar, Lamb’s test, and a soil–structure interaction problem with PML-truncated semi-infinite heterogeneous media are investigated to illustrate the efficiency of the proposed PML in terms of accuracy and CPU time.

Numerical simulation and absorbing boundary conditions for wave propagation in a semi-infinite media with a linear isotropic hardening plastic model

The theory of plastic wave propagation finds its applications in many geotechnical earthquake engineering problems. In this paper, a linear isotropic hardening plastic material is chosen to investigate the plastic wave propagation in a semi-infinite media. The dynamic response of a semi-infinite bar subjected to a triangular load is first derived. The numerical modelling scheme is then proposed and verified by the existing and derived analytical solutions. Novel artificial boundary conditions (ABCs), which are designed for transient dynamic analysis of plastic dilatational wave propagation problem in the media with a linear hardening plastic property, are developed and incorporated into the numerical scheme. Numerical examples illustrate the accuracy of the ABCs in reflecting the complex phenomenon of wave interaction, and reveal good absorption of outgoing stress waves. The study outcome provides reference for developing suitable ABCs for media with complicated nonlinear mechanical properties, and will facilitate the numerical technique in solving soil-structure seismic interaction problems.

Attenuation of waves in a viscoelastic peridynamic medium

The effect of spatial nonlocality on the decay of waves in a dissipative material is investigated. The propagation and decay of waves in a one-dimensional, viscoelastic peridynamic medium is analyzed. Both the elastic and damping terms in the material model are nonlocal. Waves produced by a source with constant amplitude applied at one end of a semi-infinite bar decay exponentially with distance from the source. The model predicts a cutoff frequency that is influenced by the nonlocal parameters. A method for computing the attenuation coefficient explicitly as a function of material properties and source frequency is presented. The theoretical results are compared with direct numerical simulations in the time domain. The relationship between the attenuation coefficient and the group velocity is derived. It is shown that in the limit of long waves (or small peridynamic horizon), Stokes’ law of sound attenuation is recovered.

Hybrid asynchronous absorbing layers based on Kosloff damping for seismic wave propagation in unbounded domains

This paper presents a novel approach for modeling infinite media, called Hybrid (different time integrators) Asynchronous (different time steps) Kosloff Absorbing Layers with Increasing Damping (HA-Kosloff ALID). By using strong forms of wave propagation in Kosloff media, its design equation is derived as well as optimal conditions between physical and absorbing domains. Explicit/Implicit co-simulation is adopted to reduce computation time. Examples of semi-infinite bar and Lamb’s test are implemented to illustrate the efficiency of our approach in terms of accuracy and CPU time, in comparison to Rayleigh ALID and PML. It turns out to be efficient and convenient for modeling unbounded domain.

Propagation of the nonlinear plastic stress waves in semi-infinite bar

This paper presents the propagation longitudinal nonlinear plastic stress in thin semi-infinite rod or in wire. The rod is characterized by a nonlinear strain hardening model within the scope a plastic strain. The modulus of strain hardening is a decreasing function of the strain. The frontal bar end is suddenly launching to the velocity V, and subsequently moves with this one. General solution of this boundary value problem of the Lagrangian coordinate (material description) and of the Eulerian one (spatial description) has been presented. There has been carried out the physical interpretation of the obtained results by means of Lagrangian and Eulerian methods. The results of this paper may be utilized in scientific researches and in engineering practice.

Calculation of Temperature Fields in a Composite Semi-infinite Body Taking Into Account the Generalized Fourier Law

Temperature fields in extended solids and the solids that have symmetry can be described with a high degree of accuracy by one-dimensional heat conduction equation. In this scientific paper the half-space with the layer is simulated using one-dimensional compound bar and consideration is given to the problem of calculation of temperature fields in the compound semi-infinite bar the finite and semi-infinite parts of which have different thermal physical parameters. The heat conduction equation with the fractional derivative in time that takes into account the nonlocality of thermal processes in time is taken as a mathematical model. Lateral surfaces of the compound bar are assumed to be heat isolated and one of the ends of its restricted part maintains the constant temperature. It is assumed that the ideal thermal contact is available at the boundary of bounded and semi-infinite parts of the bar. The solution of initial value-boundary value problem is based on the use of direct and inverse Laplace transformations. The representation of obtained solution in images presents some difficulties. This paper describes the asymptotic analysis of the solution provided that the orders of fractional derivatives in both parts are equal and the variable s of Laplace transformations tends to infinity. In this case the inverse Laplace transformation was obtained using the so-called Mainardi functions. Based on tauberian theorems we can state that the obtained solution can only be used for short times and it is represented as definite integrals of Mainardi functions or generally speaking as generalized power series. In this form it can easily be used for engineering calculations.

Lakshmi - Manoj Generalized Yang-Fourier Transforms to Heat-Conduction in a Semi-Infinite Fractal Bar

In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary or partial differential equations and other relative topics is active and extensive around the world. In the past few years, the increase of the subject is witnessed by hundreds of research papers, several monographs, and many international conferences.The purpose of present paper to solve 1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Manoj Generalized Yang-Fourier transforms method.

Wave propagation in a rectangular bar, exposed to the impact tangential forces

In modern technology, there are more and more cases of action of impact loads on work items of buildings, machines and constructions, therefore, the strength calculations of these items under the dynamic effects become important. To solve the problems, arising from the dynamic effects, it is necessary to use the continuum mechanics methods and, in particular in many cases, the methods of the dynamic theory of elasticity. Analytical studies allow to find the exact problem solutions, which is very important, because the exact solutions allow to estimate the main features of the solution in general - the nature and the extent of influence of various set parameters on it. On the other hand, exact solutions are always reference and needed, in particular, for developing numerical methods for more complex cases. This work is a continuation of the works of N. Rassoulova, G. Shamilova, dedicated to studying the propagation of unsteady waves in the prisms of rectangular cross section. Approach to solving this problem differs markedly from all previous issues of the dynamics of rectangular prisms, which mainly investigated their dispersion characteristics. This paper deals with studying the process of propagation of unsteady waves in semi-infinite rectangular bars, exposed to impact shear forces on the face platform. System of exact three-dimensional motion equations of an isotropic elastic body is used. Applying a peculiar integration method, previously developed by the authors of this paper, exact analytic solutions to the posed problem for the final time value are found. The results can be implemented in the production in designing special constructions, where there are impulsive effects on them.

Approximate Analytic Solutions of Transient Nonlinear Heat Conduction with Temperature-Dependent Thermal Diffusivity

A new approach for generating approximate analytic solutions of transient nonlinear heat conduction problems is presented. It is based on an effective combination of Lie symmetry method, homotopy perturbation method, finite element method, and simulation based error reduction techniques. Implementation of the proposed approach is demonstrated by applying it to determine approximate analytic solutions of real life problems consisting of transient nonlinear heat conduction in semi-infinite bars made of stainless steel AISI 304 and mild steel. The results from the approximate analytical solutions and the numerical solution are compared indicating good agreement.

Longitudinal Impact of Semi-Infinite Elastic Bars: Plane Any Time Solution

Using the Sobolev–Smirnov method, we have found the exact analytical solution of a longitudinal impact of semi-infinite plane elastic bars for any time after the impact. After collision, there are loading waves from contact surfaces of bars and unloading waves from lateral surfaces. Then the unloading waves reach the opposite surface of the bars and create the reflected loading waves. These loading waves reach the other surface of the bars and generate new unloading waves. The number of waves grows exponentially. The sum of waves tends to the wave of the one-dimensional approximation.

Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar

This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions. We apply a random Fourier sine and cosine transform mean square approach. Random operational mean square calculus is developed for the introduced transforms. Using previous results about random ordinary differential equations, a closed form solution stochastic process is firstly obtained. Then, expectation and variance are computed. Illustrative numerical examples are included.

The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar

1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Yang-Fourier transforms method. The simplicity and the accuracy of the method are discussed.

Longitudinal Impact of Semi-Infinite Elastic Bars: Plane Short Time Solution

Using the Sobolev-Smirnov method we find the exact analytical solution of the longitudinal impact of semi-infinite plane elastic bars. The solution allows us to investigate wave propagations shortly after the impact. The obtained results allow us to compare the exact and approximate solutions of impact problems. We show that approximate solutions are wrong in a short time. According to the exact solution five waves are created in the first time after impact: one longitudinal loading wave and two unloading longitudinal waves from the lateral surfaces of the bars and two unloading transverse waves, also from the surfaces. We find singular points of the solution. It can be used for other applications such as any impact of two elastic bodies since the same waves arise and the same singular points exist at the impact. One of the singular points moves along surfaces with the speed of Rayleigh waves. At this point the bar surface is perpendicular to the axis of the bars. Such a point arises at any impact. As the earth surface is also perpendicular to the direction of propagation of seismic waves, Rayleigh waves are especially destructive at earthquakes.

Localized vibrational modes in bars and plates

We consider the localized vibrational modes that can exist at the edge of a semi-infinite plate and at the end of a semi-infinite bar of small thickness. It is known that for certain special values of Poisson’s ratio σ these modes are perfectly localized, are uncoupled to bulk modes, and thus do not lose energy by acoustic radiation. We show that for other values of σ it is possible to modify the shape of the end of the plate or bar in a way such that a perfectly localized edge mode is formed. Finally, we discuss the effect of this localization phenomenon on the vibrational modes of plates and bars of finite length.

Effect of elastic target material on Taylor model

A theoretical model is proposed for a semi-infinite elastic bar struck axially by a flat-ended cylindrical projectile. The model is actually developed from the classical Taylor model except that the target is a semi-infinite bar with elastic behavior being considered. Particular attention is paid to the influence due to elastic wave propagation in the target bar on the energy partitioning between the projectile and target, which may result in the final length of the cylinder significantly different from the predictions of the classical Taylor model. The theoretical model is verified by numerical simulations, and the effects of several key non-dimensional parameters on the residual deformation and energy dissipation are discussed in detail. It is shown that the elastic effect of the target bar plays an important role in the prediction of plastic deformation of the cylindrical projectile.

Wave propagation in semi-infinite bar with random imperfections of density and elasticity module

Mathematical modeling and properties of a linear longitudinal wave propagating in a slender bar with random imperfections of material density and Young modulus of elasticity is discussed. Fluctuation components of material properties are considered as continuous stochastic functions of the length coordinate. Two types of fluctuation and their influence on response properties have been investigated, in particular the delta correlated and a diffusion-type processes. Investigation itself is based on Markov processes and corresponding Fokker–Planck–Kolmogorov equation. The stochastic moments closure as a solution method has been used. Many effects due to the stochastic nature of the problem have been detected. Along the bar a drop of the mean value of the response with the simultaneous increase of the response variance have been observed. This effect does not represent any conventional damping, but a gradual drop of the deterministic and an increase of the stochastic components of the overall response. The rate of the response indeterminacy increases with the increase of the length coordinate. Increasing values of material imperfection variances and the rising excitation frequency can lead to a critical state when the length of the propagating wave is comparable with the correlation length of imperfections. This state will manifest itself as a radical change of the response character. The problem will pass beyond the boundaries of stochastic mechanics and lose its physical meaning. Similar effects can be observed in the FEM analysis, where there is also a certain permissible upper boundary of the excitation frequency corresponding with the size and type of the element used.