Performance evaluation of accelerometers used for penetration experiments

We present a Hopkinson bar technique to evaluate the performance of accelerometers that measure large amplitude pulses, such as those experienced during projectile penetration tests. An aluminum striker bar impacts a thin Plexiglas or copper disk placed on the impact surface of an aluminum incident bar. The Plexiglas or copper disk pulse shaper produces a nondispersive stress wave that propagates in the aluminum incident bar and eventually interacts with a tungsten disk at the end of the bar. A quartz stress gage is placed between the aluminum bar and tungsten disk, and an accelerometer is mounted to the free end of the tungsten disk. An analytical model shows that the rise time of the incident stress pulse in the aluminum bar is long enough and the tungsten disk length is short enough that the response of the tungsten disk can be accurately approximated as rigid-body motion. We measure stress at the aluminum bar-tungsten disk interface with the quartz gage and we calculate rigid-body acceleration of the tungsten disk from Newton's Second Law and the stress gage data. In addition, we measure strain-time at two locations on the aluminum incident bar to show that the incident strain pulse is nondispersive and we calculate rigid-body acceleration of the tungsten disk from a model that uses this strain-time data. Thus, we can compare accelerations measured with the accelerometer and accelerations calculated with models that use stress gage and strain gage measurements. We show that all three acceleration-time pulses are in very close agreement for acceleration amplitudes to about 20,000 G.     

Effective medium method in the problem of axial elastic shear wave propagation through fiber composites

The effective medium method (EMM) is applied to the solution of the problem of monochromatic elastic shear wave propagation through matrix composite materials reinforced with cylindrical unidirected fibers. The dispersion equations for the wave numbers of the mean wave field in such composites are derived using two different versions of the EMM. Asymptotic solutions of these equations in the long and short wave regions are found in closed analytical forms. Numerical solutions of the dispersion equations are constructed in a wide region of frequencies of the incident field that covers long, middle and short wave regions of the mean wave field. Velocities and attenuation factors of the mean wave fields in the composites obtained by different versions of the EMM are compared for various volume concentrations and properties of the inclusions. The main discrepancies in the predictions of different versions of the EMM are indicated, analyzed and discussed.     

Elastic waves in truncated cones

An experimental investigation of elastic waves produced by the axial collision of strikers with truncated 2024 aluminum cones with apex angles of 0.48, 5.38, 20, and 30 deg was performed. Wave propagation was initiated at the small end of all four cones and at the large end of the 0.48-deg and 5.38-deg cones. The striker consisted of a 1/2-in.-diam steel ball or a soft phenol-impregnated fiber cylinder. In most cases, impact was caused by firing the striker from an air gun at approximately 1300 ips; in an additional series of tests, a steel ball was dropped on the cone. The metamorphosis of the pulse at the surface of the target was recorded using both foil and semiconductor resistance strain gages. Data were obtained for periods ranging from 200 to 500 μsec; this permitted the observation of several reflections from the ends of the specimen. In several instances, cylindrical aluminum rods were glued to the cone to form a composite target; this permitted observation of the initial pulse incident on the conical section both from surface strain gage and sandwiched crystal records. Studies were also conduced to ascertain the stress distribution across the base of the 20-deg cone. Initial pulse records were employed to predict the surface response in the target using the one-dimensional equation of elastic wave propagation in a cone of infinite length. Reasonable agreement between the data and the results of calculations based on the analysis was obtained.     

Analysis of vibrating-wire stress gage in soft rock

An analysis for the in situ elastic response of a vibrating-wire stress gage in soft rock (E<10⁵ Mpa) has been accomplished with the use of the finite-element method and approximate analytical procedures. The vibrating-wire stress gage is not a true stressmeter in that it is somewhat dependent on rock-modulus changes. Conventional procedures for calibration include the assumption that this dependency is linear with respect to rock modulus. In this analysis, a nonlinear relationship is demonstrated and an approximate mathematical expression is derived for it. Good agreement is obtained between theoretical calibration factors and test results on laboratory-scale samples with the stressmeter.     

Wave dispersion in randomly layered materials

Wave scattering in materials composed of two kinds of alternating layers with different elastic properties and randomly distributed thicknesses has been modeled. The general form of the dispersion equation is derived for the unbounded layered medium. It defines two basic macroscopic characteristics of the scattered wave: phase velocity and attenuation, which are explicit functions of wave frequency and microscopic parameters of the system: acoustic properties of the layers and stochastic characteristics of their thickness distributions. The analytical expressions are derived for three special cases: for long waves; for a periodic medium composed of layers with constant thicknesses and for random medium with uniform distribution of layer thicknesses. Special attention is paid to the analysis of the frequency dependence of the wave parameters. It was shown that the predictions of the model for long waves and for periodic medium are compatible with the results obtained in the literature.Moreover, comparison of theoretical results for frequency dependent wave parameters with numerical simulations of pulse transmission through the slab of the randomly layered medium shows good qualitative and quantitative agreement in wide frequency range.     

A theory of pulse propagation in anisotropic elastic solids

A theory is described for the propagation of pulses in anisotropic elastic media. The pulse is initially defined by a harmonically modulated Gaussian envelope. As it propagates the pulse remains Gaussian, its spatial form characterized by a complex-valued envelope tensor. The center of the pulse follows the ray path defined by the initial velocity direction of the pulse. Relatively simple expressions are presented for the evolution of the amplitude and phase of the pulse in terms of the wave velocity, the phase slowness and unit displacement vectors. The spreading of the pulse is characterized by a spreading matrix. Explicit equations are given for this matrix in a transversely isotropic material. The rate of spreading can vary considerably, depending upon the direction of propagation. New reflected and transmitted pulses are created when a pulse strikes an interface of material discontinuity. Relations are given for the new envelope tensors in terms of the incident pulse parameters. The theory provides a convenient method to describe the evolution and change of shape of an ultrasonic pulse as it traverses a piecewise homogeneous solid. Numerical simulations are presented for pulses in a strongly anisotropic fiber reinforced composite.     

Poincaré oscillations and geostrophic adjustment in a rotating paraboloid

Free liquid oscillations (Poincaré oscillations) in a rotating paraboloid are investigated theoretically and experimentally. Within the framework of shallow-water theory, with account for the centrifugal force, expressions for the free oscillation frequencies are obtained and corrections to the frequencies related with the finiteness of the liquid depth are found. It is shown that in the rotating liquid, apart from the wave modes of free oscillations, a stationary vortex mode is also generated, that is, a process of geostrophic adjustment takes place. Solutions of the shallow-water equations which describe the wave dynamics of the adjustment process are presented. In the experiments performed the wave and vortex modes were excited by removing a previously immersed hemisphere from the central part of the paraboloid. Good agreement between theory and experiment was obtained.     

Elastic wave scattering from a penny-shaped interface crack in coated materials

This paper is concerned with the elastic wave scattering induced by a penny-shaped interface crack in coated materials. Using the integral transform, the problem of wave scattering is reduced to a set of singular integral equations in matrix form. The singular integral equations are solved by the asymptotic analysis and contour integral technique, and the expressions for the stress and displacement as well as the dynamic stress intensity factors (SIFs) are obtained. Using numerical analysis, this approach is verified by the finite element method (FEM), and the numerical results agree well with the theoretical results. For various crack sizes and material combinations, the relations between the SIFs and the incident frequency are analyzed, and the amplitudes of the crack opening displacements (CODs) are plotted versus incident wavenumber. The investigation provides a theoretical basis for the dynamic failure analysis and nondestructive evaluation of coated materials.     

Short crested wave–current forces around a large vertical circular cylinder

An analytical solution for the diffraction of short crested incident wave along positive x-axis direction on a large circular cylinder with uniform current is derived. The important influences of currents on wave frequency, water run-up, wave force, inertia and drag coefficients on the cylinder profiles are investigated for short-crested incident wave. Based on the numerical results, we find wave frequency of short crested wave system is affected by incident angle and the strength of the currents. The wave frequency increases or decreases with increasing current speed following or opposing wave propagating direction. It shows that the effects of current speeds, current directions on water run-up on the circular cylinder with different radius for different wave numbers are very conspicuous when the incident wave changes from long crested plane waves to short-crested waves. With the increase of current speed, the water run-up on the cylinder becomes more and more high, and will exceed that of long crested plane wave and short crested wave case without currents even though the current speed is small. The total wave loads, inertia coefficient and drag coefficient exerted on a cylinder with currents would be larger compared to the wave loads exerted pure short-crested waves. Therefore, ocean engineers should consider the short crested wave–current load on marine constructs carefully.     

Analysis of elastic waves from two-point strain measurement

Provided that the one-dimensional wave equation applies, strain measurement at two sections of a linearly elastic cylindrical rod makes it possible to determine a number of important quantities at an arbitrary section of the rod; for example, strain, particle velocity and power transmission. The equations needed are derived, and the design of an analogue real-time analyzer is presented. The influence of some principal sources of error is analyzed and it is shown that it should be possible to perform accurate evaluation (errors less than a few percent) during a time interval which is not very long compared to the travelling time for a wave between the two gage positions. Comparisons are made between direct measurement and digital evaluation of strain, and between digital and analogue evaluation of particle velocity and power transmission. The discrepancies are typically less than ten percent during a time interval of 20 travelling times between gages. Although these results do not represent what is achievable, the accuracy is sufficient in several applications and demonstrates the feasibility of the method used.     

ACOUSTICAL TRANSMISSION LINE MODEL FOR ULTRASONIC TRANSDUCERS FOR WIDE-BANDWIDTH APPLICATION

An improved model for ultrasonic transducers is proposed.By considering only the first symmetric mode,each layer is represented as an acoustical transmission line in modeling of bulk wave transducers.In imaging applications,wide bandwidth and short time duration are required.The approach we have used consists of impedance matching the front face of the piezoelectric transducer to the propagating medium with a quarter wavelength impedance matching layer and inserting an unmatching quarter wavelength acoustical layer between the rear face and backing material.A heavy backing would degrade the wide-band phenomena,but show a time duration shorter than 0.5 μs for imaging applications.PSPICE code of the controlled source model is implemented to precisely predict the performance of the matched transducers such as impedance,insertion loss,bandwidth and duration of the impulse response.Good agreement between the simulation and experimental results has been achieved.     

Buckling of long thick-walled circular cylindrical shells of isotropic incompressible hyperelastic materials under uniform external pressure

For isotropic incompressible hyperelastic materials, the problem of determining the critical external pressure at which a long thick-walled circular cylindrical shell will buckle involves solving a fourth-order system of highly non-homogeneous, ordinary differential equations. Closed-form solutions of this system are derived here for plane-strain conditions and for the particular case of the Varga material. These solutions are used to derive the buckling criterion and numerical values are obtained for the resulting critical pressures. They are found to be in good agreement with existing theoretical and experimental results for the neo-Hookean material.     

Evaluation of tidal residual currents in a wide estuary,using a finite element method

From the linearized, time-independent, constant depth, shallow water tidal equations in an f-plane for a two-layer estuary, two independent modal Helmholtz equations are derived. These modal equations are solved using a fifth-degree finite element technique. The first and second space derivatives of the complex modal tidal elevations, and thus the modal currents and their first derivatives, are evaluated directly from the solution at each node of the finite element mesh. The Stokes drift, which is the major part of the residual tidal flow, is evaluated from these nodal values of the currents and their derivatives. Good agreement is obtained with the exact analytical solution for a wedge-shaped estuary with a wedge angle of π/3, using a mesh of 64 equilateral triangles with sides approximately 1/10 of the wavelength 2πC₂/σ of a Kelvin wave solution for the short-wavelength mode.     

Scattering coefficients for a sphere in a visco-acoustic medium for arbitrary partial wave order

Analytical solutions are reported for the scattering coefficients of a solid elastic sphere suspended in a viscous fluid for arbitrary partial wave order. Expressions are derived for incident compressional and shear wave modes, taking into account the viscosity of the surrounding fluid and resultant wave mode conversion. The long compressional wavelength limit is employed to simplify the derivation, whereas no restriction is placed on the shear wavelength in the fluid compared to the particle dimension. The analytical approximations are compared with numerical results obtained from matrix inversion of the boundary equations and agree within the validity domain of the solutions.     

NONLINEAR FLUID FORCES IN CYLINDRICAL SQUEEZE FILMS. PART I: SHORT AND LONG LENGTHS

Using three approximation methods, nonlinear models have been derived for short and long cylindrical squeeze films with arbitrary inner cylinder motions. Elliptical and parabolic velocity profiles are employed in the derivation in order to determine the effects of the choice of velocity profile. The only differences in the final squeeze film equations, due to the three approximation methods and the two velocity profiles, are in the four constant coefficients. Each term in the squeeze film equations is a nonlinear function of cylinder position. Comparing the present nonlinear expressions with existing models for short cylindrical squeeze films shows that the force terms are either exactly the same or have the same trends with instantaneous eccentricity values. For long cylindrical squeeze films, the present expressions have some force terms which are essentially the same as in other studies, while other force terms show variations with position which are very different from a previously published study.     

Transverse radiation pressure and torque exerted by a plane electromagnetic wave on a rotating circular cylinder of isotropic material

Relativistically correct expressions are derived for the time-averaged total force and torque acting on a rotating conducting circular cylinder, immersed in an incident plane electromagnetic wave. With the help of Maxwell's stress tensor analytical expressions can be obtained for the time-averaged force and torque.     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

Kolsky Bar Wave Separation Using a Photon Doppler Velocimeter

A method to permit wave separation with a Kolsky bar is described. A photon Doppler velocimeter (PDV) is used to measure particle velocity at the location of each strain-gage. These measurements are used with the measured strain to separate wave-trains that in general exist in each bar (for example, an incident and reflected pulse) even when they are superimposed at the gage location. This can extend the duration of the experiment and permit more freedom in the types of loadings that can be applied to a specimen. It was found that the PDV measurement of particle velocity often contains a significant component due to bending waves. A method to account for bending is described but requires multiple PDV measurements at each gage position.     

Transient waves in composite materials

A continuum theory for transient wave propagation in three-dimensional composite materials is given. The derived model provides a set of governing equations for the prediction of dynamic response of elastic composites to impulsive loadings. Pulse propagation normal to the direction of layering in periodically bilaminated media, and normal to the fiber direction in unidirectional long-fiber composites are obtained as special cases. The dynamic response of the composite is determined solely from the materials properties of the constituents (assumed in general to be orthotropic) and their geometrical dimensions. The predicted propagating transient waves are checked with exact solutions for impacted laminated composites, and with measured data for a fiber-reinforced material. Applications are given for pulse propagation in particulate composites and in tri-othogonally fiber-reinforced materials.