Waves In Elastic Bars Research Articles

Waves in elastic bars under axial constant velocity loading

Elastic bars serve as primary mechanical members, spanning from atomic to macroscopic scales. The development of theoretical frameworks for elastic bars is crucial for guiding a wide array of industrial and everyday applications. In this study, we comprehensively investigate wave propagation in elastic bars subjected to tension and compression under constant velocity loading conditions. First, axial waves were studied by analytically solving the axial wave equation with damping. Subsequently, we employed this theoretical framework in atomistic simulations to measure loading-rate effects and thus determine the conditions for distinguishing quasi-static and dynamic loading. Given the coupled compressive axial waves, deflection waves were analyzed to predict dynamic instability. Our findings revealed that a continuous increase in impact velocity may not necessarily result in a larger impact force. This study offers fresh insights into the mechanical theory on the dynamic behaviour of bars across diverse scales (atomic to macroscopic). These insights are invaluable for determining suitable loadings in atomistic simulations, and provide guidelines for engineering impact applications.

Normal waves in elastic bars of rectangular cross section

This paper addresses a theoretical study of guided normal waves in elastic isotropic bars of rectangular cross-section by an analytical superposition method. Dispersion properties of propagating and evanescent modes for four families are analyzed in detail at various geometric and physical parameters of the bar. A comparison of the obtained results with the well-known properties for waves in infinite plates and circular cylinders is provided. The complicated structure of dispersion spectra is explained. High-frequency limiting values for phase and group velocities of normal waves are established for the first time. Calculated data agree well with the available experimental results.

Wave dispersion and attenuation in viscoelastic polymeric bars: Analysing the effect of lateral inertia

Elementary one-dimensional wave theory is often used to describe the propagation of longitudinal stress waves in polymer rods. More accurate solutions are available, but they are mathematically difficult. A new wave equation is derived for long polymeric rods in this paper. The material properties are modelled as a Maxwell viscoelastic material acting in parallel with an elastic material. Lateral motions of the rod that result from the Poisson effect are accounted for using a new concept called the “effective density”. The effects of both the material properties and the diameter of the bar on dispersion and attenuation coefficients are highlighted. The new wave theory simplifies to the one-dimensional solution for waves in polymer rods if the Poisson ratio is set to zero. The predictions simplify to Love's equation for stress waves in elastic bars when rate dependency is removed from the material model.

Elastic waves in Timoshenko beams: the ‘lost and found’ of an eigenmode

This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler–Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful ‘branch’ of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution and (ii) a singular solution ; ϵ , ω * and k * are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ϵ . The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.

Higher-order rod approximations for the propagation of longitudinal stress waves in elastic bars

The elementary one-dimensional wave theory is often used to describe the propagation of longitudinal stress waves along a slender bar. However, for relatively large diameter bars and high-frequency waves, geometrical wave dispersion due to lateral inertia occurs, rendering this single-mode theory inaccurate. In this paper, an approximate four-mode rod equation for a circular bar that takes wave dispersion into account is presented. Little difference is observed between the dispersion curves of the four-mode equation and the Pochhammer–Chree equation that gives the exact propagation coefficient for an infinite circular bar. The advantage of the former is that its solution can be computed directly whereas the latter requires an iterative procedure. Also presented are different orders of rod approximation equations derived from the Pochhammer equation, each of which can also be solved directly. Applications of the more accurate rod equations include correcting for wave dispersion in a split Hopkinson pressure bar (SHPB) test, and more accurately determining the frequency-dependent elastic modulus of a viscoelastic bar from an experimentally measured propagation coefficient.

The spalling of long bars as a reliable method of measuring the dynamic tensile strength of ceramics

An analysis of the spall bar test as a reliable method of determining the dynamic tensile strength of brittle materials is presented. The method is based on the propagation and reflection of elastic waves in bars. Failure occurs when compressive waves are reflected into tensile ones on reaching a free end. The study analyses the hypotheses needed to obtain the true tensile strength with this experimental technique, referring to the requirements of the material and of the experimental procedure. The analysis is complemented with numerical simulations of the testing procedure. The correct way to determine the true dynamic tensile strength of ceramic materials is outlined. Finally, the results of tests of some ceramic materials, three different aluminium oxides, an alumina reinforced with zirconia, silicon carbide and boron carbide are presented.

Plastic wave propagation in Hopkinson bar - revisited

The subject of this paper is an analysis of the experimental Hopkinson bar technique when such a device consists of a short tensile or shearing specimen surrounded by two very long elastic bars [1]. Unlike the commonly applied by-pass analysis which attempts to draw conclusions from the behaviour of elastic bars, we attempt to take into account real plastic waves inside the specimen with several hundreds of reflections. A quasi rate-independent as well as a more general, rate-dependent tensor function model for AISI 316H calibrated in [19] are applied. Some special slightly perturbed elastic incident and reflected waves in elastic bars served to simulate the starting solutions. The numerical results have shown a good agreement with experimentally observed homogeneous strain state throughout the specimen during the process. Lindholm's procedure for finding specimen stress and strain by such a by-pass procedure is criticized.

APPROXIMATE DETERMINATION OF DISPERSION RELATIONS AND DISPLACEMENT FIELDS ASSOCIATED WITH ELASTIC WAVES IN BARS: Method Based on Matrix Formulation of Hamiltons Principle

A method for approximate determination of dispersion relations and displacement fields associated with elastic waves in long prismatic bars is established. The method is based on a matrix formulation of Hamilton's principle and utilizes co-ordinate functions to model the displacements in the bar. The method, which involves the solution of an eigenvalue problem, is suitable for use in a mathematical toolbox. It is applied to wave propagation in bars with square and circular cross-sections. The results obtained agree well with the known exact and approximate solutions. The convergence rate is illustrated by systematically changing the order of the co-ordinate functions.

An Equal Area Rule for Dissipative Kinetics of Propagating Strain Discontinuities

Travelling phase boundaries and shock waves in elastic bars with fairly general nonlinear stress-strain relations are investigated. The stress can be either a nonmonotone function of strain with a local maximum followed by a local minimum, or it may be monotone, but its curvature is always assumed to change sign. The effects of adding linear strain gradient but nonlinear strain rate effects to the stress are considered, with emphasis on the kinetics of travelling strain discontinuities. For a special nonlinearity in the viscosity this leads to an explicit additional jump condition for phase boundaries, valid for a general class of stress-strain laws. This is interpreted as a generalized equal area rule for dynamical phase transition, which reduces to the Maxwell rule in the static case. As a consequence, the thermodynamic force acting on phase boundaries (driving traction) is delivered through a kinetic relation as a function of their speed. Shock waves are restricted by physically meaningful inequalities.

Dispersion of elastic waves in bars with polygonal cross section

This paper studies harmonic wave propagation in an infinite elastic bar of polygonal cross section with stress-free surface. The frequency equations for longitudinal, torsional, and flexural waves have been obtained by making use of the Fourier expansion collocation method which has been developed by the author on the vibration and dynamic response problems of membranes and plates.

Dispersion Relation of Elastic Waves in Bars of Rectangular Cross Section

A formal solution for travelling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented. The solution is obtained by crosswise superposition of two single series, each term of which gives zero shearing stresses on the lateral surfaces. Frequency equations, which define the dispersion relations of elastic waves, are given by infinite determinants. Based on the approximate frequency equations, dispersion relations for a longitudinal mode, a torsional mode, and bending modes were calculated with the aid of a digital computer. Numerical results were compared with those given by other authers.

Extensional Waves in Elastic Bars of Rectangular Cross Section

A one-dimensional approximate theory is derived that accounts for the first eight modes of propagation of extensional waves in uniform, isotropic, linear elastic bars of rectangular cross section. The theory provides an optimal quadratic approximation to the lateral spatial dependence of the three-dimensional bar response. Applicable to motions in bars of arbitrary rectangular section, it is possible to predict bar responses that are neither plane stress nor plane strain, but are transitional between the two. The relation of bar to plate theory is established, and an identification of bar and plate modes is proposed. Discussion of dispersion of harmonic waves is limited to bars of square section. Typical spectra are given, and their variation with Poisson's ratio is established.

Vibrations and Waves in Elastic Bars of Rectangular Cross Section

An exact solution of the equations of elasticity is found for a family of modes of vibration, or waves, in an infinite bar of rectangular cross section for certain ratios of width to depth. The solution is composed of coupled dilatational and equivoluminal waves and the four faces of the bar are free of traction.