Pochhammer Chree Research Articles

High accuracy solutions for the Pochhammer–Chree equation in elastic media

This study investigates the nonlinear Pochhammer–Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He’s variational iteration method (VIM). The Pochhammer–Chree equation’s relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He’s VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering.

Bifurcation analysis, quasi-periodic and chaotic behavior of generalized Pochhammer-Chree equation

We take into account the bifurcation analysis of the generalized Pochhammer-Chree (PC) equation that describes the dynamics of several systems in science and engineering. The considered model is changed into a planar dynamical system by applying the Galilean transformation. The phase portraits are plotted by considering suitable values of the bifurcation parameters. The considered model is solved using the RK method to compute the supernonlinear and nonlinear wave solutions. All phase portraits and wave solutions are depicted in the phase plane by simply fixing the relevant parameters values. The equilibrium points are obtained, and the same are classified. Moreover, sensitive analysis for different initial value problems is applied to analyze the quasiperiodic, chaotic behavior and time series after introducing an extrinsic periodic perturbation term. In addition, the Lyapunov characteristic exponents, Poincare section and bifurcation diagrams are also discussed to examine the chaotic pattern of the model. Numerical simulation results show that changing the frequencies and amplitude values impacts the dynamical features of the considered model.

Determining longitudinal- and shear-wave velocities in Japanese radish, watermelon and potato

Longitudinal- and shear-wave velocities were determined for Japanese radish, watermelon and potato. Pulsed waves propagating in cylindrically shaped specimens were detected, and their phase differences obtained in specimens of different lengths yielded phase velocities in the frequency range 0.4–8 kHz at 27 kHz. The Pochhammer–Chree dispersion relation was applied to analyze the fundamental and second modes of the measured velocities, which provided the longitudinal- and shear-wave velocities in an infinite medium. Longitudinal-wave velocities of 202, 82 and 571 m s−1 and shear-wave velocities of 61, 36 and 80 m s–1 for Japanese radish, watermelon and potato, respectively, were obtained. Furthermore, the Young’s modulus and Poisson’s ratio were determined. The drying effect of the velocities obtained in the Japanese radish suggested that gas porosity substantially influenced the velocities, resulting in smaller values of the longitudinal-wave velocity compared with that in water.

Symmetry analysis, optimal subalgebra, quasi-self-adjointness condition with conservation laws and analytical solutions for the ($$1+1$$)-dimensional Pochhammer–Chree model in longitudinal wave propagation

Symmetry analysis, optimal subalgebra, quasi-self-adjointness condition with conservation laws and analytical solutions for the ($$1+1$$)-dimensional Pochhammer–Chree model in longitudinal wave propagation

Double reductions and traveling wave structures of the generalized Pochhammer–Chree equation

Symmetry methods are always very useful for discussing the classes of differential equation solutions. This article focuses on traveling wave structures of the generalized Pochhammer–Chree (PHC) equation. First, we will discuss Lie point symmetries of the PHC equation to classify the solutions. Then, we formulate traveling wave structures considering the reduced differential equations (DEs) by using sech method and the new extended direct algebraic (EDA) method. In the end, we will sketch some of the traveling wave structures.

Solution of the External Pochhammer–Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuum

Solution of the External Pochhammer–Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuum

A NOVEL COMPUTATIONAL APPROACH FOR FRACTAL NONLINEAR POCHHAMMER–CHREE MODEL

The purpose of this paper is to establish a novel algorithm to find the fractal solitary wave solution for fractal nonlinear Pochhammer–Chree model based on fractal variational theory, which is called variational transform method (VTM). The numerical examples are given to reveal the effectiveness and simplicity of the proposed method. Finally, some physical properties of the obtained fractal traveling wave solutions are illustrated by some graphs. The novel method sheds a new light on the fractal soliton theory.

Stresses in Pochhammer–Chree Axisymmetric Waves in the Medium-Wavelength Range

Stresses in Pochhammer–Chree Axisymmetric Waves in the Medium-Wavelength Range

The exact solutions of the conformable time fractional version of the generalized Pochhammer–Chree equation

The time-fractional version of the generalized Pochhammer–Chree equation is analyzed. In this paper, the equation is converted into an ordinary differential equation by applying certain real transformation, then the discrimination of polynomials system is used to find exact solutions depending on the fractional order derivative. The obtained solutions are graphically illustrated for different values of the fractional order derivative keeping the other parameters fixed.

Manual for Calibrating Sound Speed and Poisson’s Ratio of (Split) Hopkinson Bar via Dispersion Correction Using Excel® and Matlab® Templates

This manual presents a procedure to calibrate the one-dimensional sound speed (co) and Poisson’s ratio (ν) of a (split) Hopkinson bar using the open-source templates written in Excel® and Matlab® for dispersion correction. The Excel® template carries out the Fourier synthesis and one-time dispersion correction of a traveling elastic pulse under a given set of co and ν. The MATLAB® template performs the Fourier synthesis and iterative dispersion correction of a traveling elastic pulse for a range of co and ν sets. In the case of the iterative dispersion correction, a set of co and ν is assumed at each iteration step, and the sound speed vs. frequency (cdc vs. fdc) relationship necessary for dispersion correction is obtained under the assumed set by solving the Pochhammer–Chree equation. Subsequently, dispersion correction is carried out by using the cdc vs. fdc relationship. The co and ν values of the bar are determined in the iteration process when the dispersion-corrected pulse profiles are reasonably consistent with the measured ones at two travel distances (2103 and 4000 mm) in the bar. In the case of the experimental profile considered herein, the ν and co values were calibrated to six and four decimal places, respectively. The calibration algorithm is described with the tips for using the open-source templates, which are available online in a publicly accessible repository.

Sound Speed and Poisson’s Ratio Calibration of (Split) Hopkinson Bar via Iterative Dispersion Correction of Elastic Wave

AbstractA process of calibrating a one-dimensional sound speed (co) and Poisson’s ratio (ν) of a (split) Hopkinson bar is presented. This process consists of Fourier synthesis and iterative dispersion correction (time-domain phase shift) of the elastic pulse generated by the striker impact on a circular bar. At each iteration, a set of co and ν is assumed, and the sound speed versus frequency (cdc versus fdc) relationship under the assumed set is obtained using the Pochhammer–Chree equation solver developed herein for ground state excitation. Subsequently, each constituting wave of the overall elastic pulse is phase shifted (dispersion corrected) using the cdc–fdc relationship. The co and ν values of the bar are determined in the iteration process when the dispersion-corrected overall pulse profiles are reasonably consistent with the measured profiles at two travel distances in the bar. The observed consistency of the predicted (dispersion-corrected) wave profiles with the measured profiles is a mutually self-consistent verification of (i) the calibrated values of co and ν, and (ii) the combined theories of Fourier and Pochhammer–Chree. The contributions of the calibrated values of co and ν to contemporary bar technology are discussed, together with the physical significance of the tail part of a traveling wave according to the combined theories. A preprocessing template (in Excel®) and calibration platform (in matlab®) for the presented calibration process are openly available online in a public repository.

Ultra-efficient in-core acoustic waves in suspended core fiber for high frequency fiber-optic ultrasonic devices

High frequency acoustic waves confined inside a suspended core fiber (SCF) are revealed by numerical simulations for the first time. A standard single mode optical fiber (SMF) is modelled to explain the current research limitations. The acoustic frequency response and energy concentrated in both fibers are evaluated from 50 to 56 MHz with the 3D finite element method. The simulated dispersion curves are compared to the Pochhammer–Chree solutions. The energy in the SCF core at the resonance of 52.84 MHz is approximately 16 times higher compared to the SMF, indicating a new route for the design and fabrication of fiber-based ultrasonic devices.

Methodological aspects of testing brittle materials using the split Hopkinson bar technique

AbstractThis paper presents the features of a method for determining the dynamic properties of brittle materials using the split Hopkinson bar technique on mortar as an example. Experiments were conducted both with and without a loading pulse shaper, a deformable copper disc (pad) placed on the impacted end of the input bar. Experimental data from the experiments without the pad were processed with and without dispersion correction based on a direct solution of the Pochhammer–Chree frequency equation of pulses in the measuring bars. There were compared both the deformation graphs and mechanical characteristics of the tested material for three variants of the split Hopkinson pressure bar experiments: with and without a loading pulse shaper, and with the dispersion correction procedure for pulses in the measuring bars. The phenomena observed during these experimental schemes and the processing of obtained data are described and discussed; further, the effect of the nature of the load increase is revealed.

Pochhammer–Chree equation solver for dispersion correction of elastic waves in a (split) Hopkinson bar

A robust algorithm for solving the Bancroft version of the Pochhammer–Chree (PC) equation is developed based on the iterative root-finding process. The formulated solver not only obtains the conventional n-series solutions but also derives a new series of solutions, named m-series solutions. The n-series solutions are located on the PC function surface that relatively gradually varies in the vicinity of the roots, whereas the m-series solutions are located between two PC function surfaces with (nearly) positive and negative infinity values. The proposed solver obtains a series of sound speeds at exactly the frequencies necessary for dispersion correction, and the derived solutions are accurate to the ninth decimal place. The solver is capable of solving the PC equation up to n = 20 and m = 20 in the ranges of Poisson’s ratio ( ν) of 0.02 [Formula: see text] ν [Formula: see text] 0.48, normalised frequency ( F) of F [Formula: see text] 30, and normalised sound speed ( C) of C [Formula: see text] 300. The developed algorithm was implemented in MATLAB®, which is available in the Supplemental Material (accessible online).

Local Phase-Amplitude Joint Correction for Free Surface Velocity of Hopkinson Pressure Bar

The Hopkinson pressure bar is widely used to measure the reflected pressure of blast waves over a short distance. However, dispersion effects will occur when the elastic stress waves propagate in the pressure bar due to lateral inertia, and there will be errors between the signals obtained from the sensors and the actual loading. For the free surface velocity measured in our system, we developed a local phase-amplitude joint correction method to convert the measured velocity into the average reflected pressure of a shock wave at the impact end of the bar, considering factors such as propagation modes of the elastic wave, the frequency components’ time of arrival, velocity variation over the bar axis, and the stress–velocity relationship. Firstly, the Pochhammer–Chree frequency equation is calculated numerically, and the first to fourth orders of phase velocity, group velocity, normalized frequency, and propagation time curves of elastic wave propagation in 35CrMnSiA steel are obtained. Secondly, the phase and amplitude correction formulas for calculating average reflected pressure from center velocity are derived based on the propagation mode of the axial elastic wave in the pressure bar by analyzing the time-frequency combined spectrum obtained by short-time Fourier transform. Thirdly, a local phase-amplitude joint correction algorithm based on propagation mode is proposed in detail. The experimental tests and data analyses are carried out for eight sets of pressure bar. The results show that this method can identify the propagation mode of elastic waves in the bar intuitively and clearly. The first three orders of propagation modes are stimulated in the bar 04, while only the first order of propagation is stimulated in the other eight bars. The local phase-amplitude joint correction algorithm can avoid correcting the component of the non-axial elastic wave. The rising edge of the average stress curve on the impact surface of bar 01 and bar 04 is corrected from 4.13 μs and 4.09 μs to 2.70 μs, respectively.

Characterization of the Mechanical Behavior of a Lead Alloy, from Quasi-Static to Dynamic Loading for a Wide Range of Temperatures.

The current needs in terms of ballistic protection for armed forces require an almost constant improvement in performance to face the constantly evolving threats and scenarios. Ballistic tests are conventionally carried out in order to assess and validate the levels of protection. The challenge is to be able to set up a digital protocol and only carry out final validation tests. Indeed, the advantage of digital simulation lies in the possibility of being able to evaluate a wide variety of configurations. In order to obtain reliable results, it is necessary to use sufficiently precise material behavior models to transcribe the phenomena observed during the impact. Our study focuses on the behavior of a small caliber ammunition with a ductile core impacting personal protection. More particularly on the mechanical behavior of the lead alloy core. Thus, compression tests have been carried out on a wide range of deformation rates, from quasi-static behavior to dynamic regime, at different temperatures. The study in dynamic conditions was carried out using split Hopkinson pressure bars. Due to the material properties, the experimental device had to be adapted in order to optimize the propagation of the waves allowing to measure signals (elastic waves). These tests demonstrate the dependency of the stress with strain rate and temperature. Dynamic restoration and recrystallization phenomena, characteristic of a material deformed in its hot working area, have also been identified. The associated oscillations due to Pochhammer–Chree effect, observable on the stress–strain curves, constitute the major problem for the implementation of behavioral models. Finally, a constitutive model sensitive to strain rate and temperature is investigated for ballistic purposes.

Influence of Imperfect Position of a Striker and Input Bar on Wave Propagation in a Split Hopkinson Pressure Bar (SHPB) Setup with a Pulse-Shape Technique

The effect of using a pulse shaper technique, such as rounding a striker or applying a pulse shaper on the signals recorded with the split Hopkinson pressure bar (SHPB) technique, when the striker and the input bar are in an imperfect position, was investigated. Two of the most common cases have been analyzed: an offset of the symmetry axes of the striker and the input bar; and an inclination angle between the striker and the input bar. LS-Dyna software was used to examine this problem numerically. The inclination angle imperfection has a significant impact on signal disturbances, whereas the use of a rounded striker significantly affects the limitation of the vibration flexural modes. In all considered cases, a slight imperfection causes a reduction in the high-frequency Pochhammer–Chree oscillations.

Correction of higher mode Pochhammer–Chree dispersion in experimental blast loading measurements

Experimental measurements of blast loading using Hopkinson pressure bars are affected by dispersion which can result in the loss or distortion of important high-frequency features. Blast waves typically excite multiple modes of propagation in the bar, and full correction of dispersive effects is not currently possible as the magnitude of stress propagating in each mode is not known. In this paper we develop an algorithm for multiple mode dispersion correction based on rigorous interrogation of the results from a series of finite element analyses. First, a finite element model is validated against first-mode and higher-mode theory. The dispersion of short raised-cosine windowed pulses is then used to isolate the contribution of each propagating mode, enabling a relationship between frequency and modal apportioning of stress to be obtained for the first four propagating modes. Finally, four-mode dispersion correction is successfully applied to an experimental signal using an algorithm based on the derived relationships for modal apportioning. The four-mode results show significant improvement in the capture of high-frequency features over existing first-mode corrections, and demonstrate the potential of this method for the full correction of dispersion in experimental measurements of blast loading.

Maxima of the Stresses in the Longitudinal Pochhammer—Chree Waves

The article presents the calculation of stress fields that arise in cylindrical rods of circular cross section in the case of Pochhammer–Chree waves. Examples of calculating the fields for the four lowest wave modes in a steel rod for two cases of phase velocities are considered. The structures of the isostat of the principal stresses in the longitudinal section of the rod are constructed and considered. The fields of the first principal stress and von Mises stress are constructed.

On the integrability of the generalized Pochhammer–Chree (PC) equations

In this paper, the generalized Pochhammer–Chree (PC) equations which the propagation of longitudinal deformation waves in an elastic rod is considered. By using the Weierstrass elliptic function method, we find exact traveling wave solutions. As a result, some previously known solutions are recovered, and at the same time some new ones are also given. Moreover, three-dimension and two-dimension graphics of some solutions have been plotted.