Waves In Bars Research Articles

A new higher order dynamic theory for thermoelastic bars. I: General theory

A dynamic approximate theory capable of predicting high-frequency behavior of cylindrical thermoelastic bars is developed using a new theory. The cross section of the bar has an arbitrary shape and contains an arbitrary number of holes. The approximate theory is valid for all of the deformation modes such as flexural, longitudinal, torsional, etc. The use of the new method permits one to eliminate any inconsistency which may occur between lateral boundary conditions and the distributions of displacements or temperature assumed over the cross section of the bar. Accordingly, the method enables one to correctly describe the reflections of the waves propagating along the bar. This, in turn, makes the dispersive characteristics of waves in bars predicted by the approximate theory agree with those obtained from the exact theory without having to introduce any matching coefficients into the approximate theory.

A new higher order dynamic theory for thermoelastic bars. II: Application to thermoelastic circular and rectangular bars

In order to show and illustrate the power of the theory proposed in Part I [J. Acoust. Soc. Am. 73, 1918–1922 (1983)] for the dynamic behavior of thermoelastic bars, the theory is applied to bars with circular and rectangular cross sections. In these applications, the order of the theory m is chosen to be 3. The general equations of the approximate theory governing all of the deformation modes (such as longitudinal, flexural, torsional, etc.) of circular and rectangular bars, and accomodating the thermal effects are presented. With the object of assessing the approximate theory, approximate and exact dispersion curves are compared for the flexural and longitudinal waves propagating in circular bars. A good match between these two is obtained without using any correction factors in the approximate theory. Further, experimental wave profiles are compared for longitudinal waves in rectangular bars. It is observed that the two agree quite well.

Design of rubber mandrel fiber optic hydrophones

The response of a fiber optic pressure sensor consisting of a long fiber waveguide wrapped azimuthally on a rubber cylinder has been analyzed. The response of the rubber cylinder includes forced waves and free bar waves. The sensitivity of the hydrophone depends on the trace wavenumber of the incident pressure field and therefore is strongly affected by the frequency and the angle of incidence of the sound wave. In addition the temperature of the environment and the frequency of the pressure field affect dynamic properties of the rubber cylinder. Axial stiffeners stabilize the properties of the hydrophone and also reduce the influence of the trace wavenumber of the incident acoustic wave. Two types of design work well, either one with stiffening sufficient to produce a slightly subsonic bar wave, or one with stiffening sufficient to produce a bar wave speed of about four times the speed of sound.

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.

Dynamic Stress Concentrations Due to Flexural Waves in Bars with Various Bends : 2nd Report, Solutions of Bars Excited in Arbitrary Directions

This paper discusses the dynamic stress concentrations in bars with various bends. The flexural wave propagating along the bar which vibrates in an arbitrary direction is considered as an incident wave. Since the bar has bends, this incident wave induces longitudinal, torsional and flexural waves at these bends. In this paper, the elementary bar theory is applied for longitudinal and torsional waves, and the improved bar theory in which the effects of rotatory inertia and shear deformation are considered is applied for flexural waves. The analytical results for general problems are obtained, and the numerical calculations are carried out for cases of a circular arc bend, and L-bend and a Z-bend bars.

Dynamic Stress Concentrations Due to Flexural Waves in Bars with Various Bends (2nd Report, Solutions of Bars Excited in Arbitrary Directions)

Dynamic Stress Concentrations Due to Flexural Waves in Bars with Various Bends (2nd Report, Solutions of Bars Excited in Arbitrary Directions)

Elastic waves in heterogeneous bars of varying cross-section

The propagation of elastic waves in a heterogeneous bar of variable cross-sectional area is investigated via use of the method of characteristics andthe Laplace transform technique. The Young's modulus and density are assumed to be representable as either power law or exponential distributions in the axial coordinate. The transform method is used to establish an infinite number of multi-parameter solutions in closed form for either a stress, velocity or displacement type boundary condition. The numerical characteristic computations show excellent agreement when compared to the transform solutions, and are then used to obtain additional solutions not attainable by the transform method. Detailed results and conclusions for a bar of ogival cross-section are given for a wide range of inhomogeneity.

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.

Dynamical Stress Concentration due to Flexural Waves in Bars with Various Bends

This paper discusses the problems of flexural wave propagation in an infinite bar with a circular arc bend, in an infinite L-type bar with a sharp bend and in an infinite Z-type bar with two bends when excited by an incident wave. A new improved theory taking into account the effects of the extension, rotatory inertia and shearing deformation is advocated. The bending moment concentration factors for these types of bars at their important cross sections are calculated and the differences of the dynamical behaviors among these bars are clarified.

An optical method for direct measurements of strain in a torsional Hopkinson-bar apparatus

A method has been developed to measure the related rotation of the flanges of a thin-walled tubular specimen during a torsion test. The method, which is based on the Moiré-fringe technique, is capable of use at the high rates of strain encountered during a Hopkinson-bar test, as well as at low rates of strain. In the application described, the specimen gauge length is very short, but the method could be used for specimens of considerably longer gauge length. Direct calibration of the system is easily carried out at low angular velocities. The method can then be used to measure directly the specimen strain during a Hopkinson-bar test, and thus to check the value derived from measuremets of torsional waves in the elastic bars. Results of such comparisons are given, and it is found that the values given by the two method agree well, the differnce being attributable largely to inaccuracies in the torque measurement. The new method permits the determination of specimen deformation during the later stages of the test when multiple wave reflections render the wave analysis iaccurate. In particular, it has been found that the specimen may be subjected to reversed plastic straining, so that the total plastic strain connot be determined from the permenent deformation at the end of the test.

Dynamic Stress Concentrations by Flexural Waves in Bars with Various Bends

Dynamic Stress Concentrations by Flexural Waves in Bars with Various Bends

Impact Strength of Materials

This book presents a compilation of information and analytical methods on the subject of the impact strength of materials. It contains ten chapters dealing with: 1, elementary one-dimensional elastic stress waves in long uniform bars due to impact; 2, applications of elementary one-dimensional stress wave theory; 3, elastic stress waves: more general considerations; 4, plasticity theory and some quasi-static analyses; 5, one dimensional elastic-plastic stress waves in bars; 6, impulsive loading of beams; 7, dynamic loading of rings and frames; 8, dynamic plastic deformation of plates; 9, plastic deformation in a semi-infinite medium due to impact; and 10, plastic deformation in plates due to impact. Two appendices contain: 1, useful conversion factors; and 2, text tables in SI units. (TRRL)

Equations of flexural ? torsional waves in thin-walled bars of open cross section

Equations of flexural ? torsional waves in thin-walled bars of open cross section

A numerical method for the analysis of longitudinal elastic–plastic stress wave propagation

AbstractA numerical method for the analysis of one‐dimensional propagation of elastic–plastic stress waves in cylindrical bars is proposed. The method is based on the division of the bar into elements of finite length and the solution of the equations of motion for these elements.Numerical results are presented for semi‐infinite bars and a finite bar fixed at one end.

On the Effect of Initial Stress on the Propagation of Flexural Waves in Elastic Rectangular Bars

The dynamics of continuous media subjected to initial stress is of great interest in several fields of applied science and technology: geophysics, oceanography, underwater acoustics, and structural design. Pioneering analytical work in this field was done by M. A. Biot. Few contributions have been made, however, to the experimental treatment of the problem of determining dynamic stress distributions within statically prestressed continuous media. This study deals with an experimental analysis of the propagation of flexural waves in prismatic elastic bars with and without prestressing. The effects of prestressing by axial tension, axial compression, and pure bending are illustrated. Comparisons are made with an approximate theory. The results are extended to bars of other materials by scaling laws.

Influence of a Surface Mass Distribution on the Motion of Bars

A method is developed for determining the modes of propagation for progressive harmonic waves in infinite rectangular elastic bars. The faces of the bar are covered with a distributed mass of negligible stiffness, as found with a layer of insulation or a damping treatment. An approximate solution is obtained by expanding the displacements as a power series in the thickness coordinates and retaining only terms through first order. The D'Alembert force due to the acceleration of the mass produces surface tractions which influence the motion and introduce coupling between the longitudinal, torsional, and flexural motions. If equal mass distributions are applied over opposite faces, the motions are qualitatively similar to the motions of a free bar. If unequal masses are added to opposite faces, the motions are more coupled than those of a free bar. The addition of mass to two adjacent faces leads to a yet higher degree of coupling.

Effect of Initial Stress on the Propagation of Flexural Waves in Elastic Rectangular Bars

The dynamics of continuous media subjected to initial stress is of great interest in several fields of applied science and technology: geophysics, oceanography, underwater acoustics, and structural design. Pioneering analytical work in this field was done by Biot. Few contributions have been made, however, to the experimental treatment of the problems of determining dynamic stress distributions within statically prestressed continuous media. The present paper deals with an experimental analysis of the propagation of flexural waves in prismatic elastic bars with and without prestressing. A complete, direct, full-field optical determination of dynamic stress distributions associated with the flexural waves is obtained by a combination of photoelastic and interferometric measurements. The effects of prestressing by axial tension, axial compression, and pure bending are illustrated. Comparisons are made with an approximate theory. The results are extended to bars of other materials by scaling laws.

Dynamic problems for elastic-plastic solids with delayed yielding

A simplified model of an elastic-plastic body with delayed yielding is discussed. The strain rate influence on the post yielding plastic resistance is neglected, the plastic deformation being assumed to follow the static stress-strain curve.The model has been used for solving two kinds of problems: propagation of longitudinal elastic-plastic waves in bars and rigid-plastic analysis of beams and plates.When a stress larger than the yield stress value is acting on the end of a bar, an elastic overstress wave spreads. In bars of finite length the reflected overstress wave strongly affects the distribution of the residual plastic strains.In rigid-plastic analysis the rigid parts of a body are considered to be elastic with elastic moduli tending to infinity. The successive formation of plastic hinges in a free beam subjected to an instantaneously applied concentrated load has been studied. A simply supported circular plate under the action of a uniform loading has been considered. The propagation of the plastic zone is governed by the stress distribution in the rigid part of the plate; at a certain moment the yield delay capacity becomes exhausted and the further analysis only slightly differs from the traditional one.

Propagation of longitudinal elastic stress waves in bars

Propagation of longitudinal elastic stress waves in bars

On the propagation of flexural waves in anisotropic bars

Propagation of flexural waves in circularly cylindrical bars of anisotropic material on the basis of the exact three-dimensional theory of elasticity of small deformations is considered, the rigor of the analysis being the same as the one of the Pochhammer-Chree theory. A system of the governing equations of motion for the cylindrical orthotropy (its axis coinciding with the axis of the bar) is derived and expressed in terms of radial, tangential and axial displacement components. For a steady-state sinusoidal flexural wave the system reduces to coupled ordinary second order differential equations with a regular singular point. Solution for a transversely isotropic is obtained by means of Frobenius series method. Upon using two or three terms of the series approximate dispersion relations for isotropic material as well as for the pyrolytic graphite type material and two other types of anisotropic materials are obtained and illustrated by graphs. Higher order modes than those already known are found.