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J. Appl. Mech. 1969;36(2):145. doi:10.1115/1.3564600.
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Abstract
Commentary by Dr. Valentin Fuster

RESEARCH PAPERS

J. Appl. Mech. 1969;36(2):146-150. doi:10.1115/1.3564601.

A supersonic low-density gas stream produced in a supersonic nozzle was passed through a circular tube in which the transition from supersonic to subsonic flow took place. Static pressure distributions along the tube (and nozzle) and impact pressure distributions across the tube at several stations were measured to determine the nature of this transition. The impact pressure distributions were used, together with the local static pressure, to infer Mach number and velocity profiles in the tube. When the pressure distributions and center-line Mach number distributions are considered together, one obtains a fairly clear picture of the processes involved in the transition from supersonic to subsonic flow at low Reynolds numbers.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):151-158. doi:10.1115/1.3564602.

Turbulent swirling flows in stationary cylindrical ducts were investigated analytically using Taylor’s modified vorticity transport theory and von Karman’s similarity hypothesis extended to consider a three-dimensional fluctuating velocity field. The resulting similarity conditions were used to formulate the expression for eddy diffusivity in the entire flow field except in a small region near the pipe wall where a mixing-length expression analogous to that assumed by Prandtl for parallel flow in channels was used. The swirl equation was solved numerically using a constant that was obtained indirectly from an experiment by Taylor, and the analytical results were compared with two different sets of experimental measurements. In both cases, the agreement between experiment and analysis was satisfactory. Some discrepancies appeared when the flow field was predominantly irrotational or in solid-body rotation: This might have been expected since, for these situations, some of the similarity conditions were indeterminate or infinite.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):159-170. doi:10.1115/1.3564603.

The pulsating flows in both infinite and finite conical nozzles were analyzed theoretically. Sinusoidal pressure disturbances were impressed at the nozzle exit for the infinite nozzle and at either the inlet or at the exit for the case of a finite nozzle. The results have been calculated in terms of mass-flux response. The parameters involved are the Mach number and the modified Strouhal number; the inlet and exit radii ratio enters as an additional parameter for a finite nozzle. The results for an infinite conical nozzle indicate that, when the frequency is low, the quasistatic relationship between the pressure and mass-flux fluctuations holds; the same was reported in reference [1]. But, as the frequency increases, the dynamic characteristics of the pulsating flow become important. And, at high frequencies, the mass-flux response is less than the quasistatic value by an amount depending on the Mach number. For a finite conical nozzle the quasistatic condition is still valid if the frequency is low. However, at higher frequencies, the dynamic behavior becomes critically dependent on the frequency expressed in terms of w, for a given nozzle geometry and exit Mach number.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):171-180. doi:10.1115/1.3564604.

Equations are developed which satisfactorily describe the change in the pressure differential with time for a closed, constant-volume system which utilizes a coiled-capillary tube and a constant-rate pump to determine gas viscosities. Viscosities are computed at transient, unsteady, and apparent-steady-state conditions. As long as the flow rates are not too high and the pressure level is not too low, the computed viscosities appear to be reliable.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):181-188. doi:10.1115/1.3564605.

Weak shock theory is used to analyze the propagation of the stress waves which are induced by nonuniform, instantaneous, internal heating of a nonlinearly elastic, semi-infinite solid. The material nonlinearity considered is caused by the increase in bulk modulus which occurs as the hydrodynamic stress component increases. Heating is assumed to occur instantaneously. Since the shock is assumed to be weak, the entropy change across it is negligible, and therefore the wave form both behind and in front of the shock is found by using a coordinate perturbation method to solve the nonlinear equations for constant entropy. This solution predicts a multivalued material state in the vicinity of the shock front without locating the front itself. The location of the shock front is found separately by using the principle of momentum conservation. If the front is then inserted at this location, a wave form is obtained for which the material state is everywhere single-valued. The results and conclusions which are presented are based on a comparison of the perturbation solution found in this paper, and a simple wave solution and on an assessment of shock strength.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):189-197. doi:10.1115/1.3564606.

The plane wave propagation in a half space due to a uniformly distributed step load of pressure and shear on the surface was first studied by Bleich and Nelson. The material in the half space was assumed to be elastic-ideally plastic. In this paper, we study the same problem for a general elastic-plastic material. The half space can be initially prestressed. The results can be extended to the case in which the loads on the surface are not necessarily step loads, but with a restricting relation between the pressure and the shear stresses.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):198-202. doi:10.1115/1.3564607.

In this paper an approximate solution is presented for the radius of contact between an elastic plate and a semi-infinite elastic half space. The plate is assumed to rest on the supporting half space without bond, and to be pressed against the elastic region by a concentrated load. In the absence of bonding no tensile stress can be transmitted across the interface between the plate and its elastic support so that contact takes place only within a circle centered about the concentrated load. Outside of this circle the plate lifts up and is no longer in contact with the elastic region.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):203-211. doi:10.1115/1.3564608.

The elastic-plastic boundaries which start at the end of a tube in wave propagation of combined longitudinal and torsional stresses are studied. The longitudinal and torsional stresses applied at the end of the tube are assumed to be continuous and the initial slope of the boundaries are determined analytically for all possible combinations of the time derivatives of stresses before and after the elastic-plastic boundaries occur. It is seen that a continuous loading into a higher yield surface at the end of the tube does not necessarily produce a continuous plastic region near the end of the tube. Also, more than one elastic and plastic region can be generated near the end of the tube even though the stresses at the end of the tube are continuously increasing to a higher yield surface. These and other unexpected results are presented in this paper.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):212-216. doi:10.1115/1.3564609.

A Lyapunov-type approach is used to establish sufficient conditions guaranteeing the asymptotic stability of a class of partial differential equations with parametric excitation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):217-220. doi:10.1115/1.3564610.

The collapse velocity of the liner in a hollow cylinder explosion is difficult to obtain experimentally and asymptotic formulas for the velocity are unavailable for such explosions. This paper presents a method for the computation of the collapse and fragment velocities by solving numerically the flow equations of an idealized flow of the explosive fumes. The results agree with available experimental data.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):221-227. doi:10.1115/1.3564611.

This paper concerns the mean-square response of a single-degree-of-freedom system to amplitude modulated random noise. The formulation is developed in terms of the frequency-response function of the system and generalized spectra of the nonstationary random excitation. Both the unit step and rectangular step functions are used for the amplitude modulation, and both white noise and noise with an exponentially decaying harmonic correlation function are considered. The time-varying mean-square response is shown not to exceed its stationary value for white noise. For correlated noise, however, it is shown that the system mean-square response may exceed its stationary value.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):228-232. doi:10.1115/1.3564612.

A displacement bounding principle for continua loaded into the plastic range is presented. Use is made of Drucker’s inequality and the existence of a maximum complimentary work path between any two stress states. The bounding principle is applied to several examples.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):233-240. doi:10.1115/1.3564613.

We consider the buckling of an ensemble of infinitely long columns, with initial deflections, resting on nonlinear elastic foundations. The initial deflections are assumed to be Gaussian stationary random functions of known autocorrelation, and the problem is solved by the method of equivalent linearization. We find that each column in the ensemble has the same buckling load that depends only on the autocorrelation of the initial deflection functions. Results are presented for columns whose initial deflection functions have an exponential-cosine autocorrelation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):241-246. doi:10.1115/1.3564615.

The problem of determining the deformation of an elastic rod of finite dimensions under a symmetrical axial loading is reduced to the solution of an infinite system of simultaneous linear equations. An approximate numerical solution is obtained for the case when the load acts uniformly over an outer annular region of the end face, and it is shown that the maximum radial displacement increases as the width of the annulus decreases. The application of the results to the problem of circumferential ridging of nuclear reactor fuel elements is discussed. It is concluded that the compression between adjacent rods probably contributes to ridge formation, but that the magnitude of the contribution cannot be determined without a knowledge of the compressive plastic stress-strain characteristics of the material.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):247-253. doi:10.1115/1.3564616.

A set of integral equations is developed for the stress rates that correspond to a given rate of loading from a given elastoplastic state in an elastic-plastic, work-hardening sandwich arch. Methods of solving these integral equations are considered. It is found that the integral-equation approach permits solutions which are as accurate as desired even when the loading program involves extensive partial unloadings. Thus the results of this paper may prove useful for checking various existing numerical techniques.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):254-260. doi:10.1115/1.3564617.

In the present paper, using an improved Reissner’s variational theorem along with Berger’s hypothesis, a set of governing equations which include the effects of transverse shear deformation and rotatory inertia is derived for the large amplitude free vibrations of plates composed of a transversely isotropic material. Applying the possibility of neglecting the rotatory inertia in primarily flexural vibration (discussed in the previous work [1]2 ), the lateral free vibrations of simply supported plates are treated in detail and the solution is compared with those of previous investigators. The free vibration of beams is studied as a special case of plates, while the small amplitude vibrations are treated as a special case of large amplitude vibrations. The numerical results show that the effect of transverse shear deformation is significant when applying to the plate constructions made of pyrolytic graphite-type materials.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):261-266. doi:10.1115/1.3564618.

Using the basic assumptions of thin-plate theory, including nonlinear terms in the von Karman sense, the governing equations of a laminated anisotropic plate are formulated. In particular, the type of plate under discussion consists of n layers of orthotropic sheets bonded together. Each layer has arbitrary thickness, elastic properties, and orientation of orthotropic axes with respect to the plate axes. The governing equations are obtained by integrating the equations of nonlinear elasticity. Inertia terms and thermal stresses are included. Closed-form solutions to the linearized equations are obtained for bending, flexural vibration, and buckling of special, but important, classes of laminates for which coupling between bending and stretching is unavoidable.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):267-270. doi:10.1115/1.3564619.

Recent simplifications of linear shell theory through consideration of transverse shear deformations and stress moments with axes normal to the shell middle surface suggest analogous approaches to the corresponding problem of nonlinear theory. As a first step in this direction consideration is given here to the classical subject of finite symmetrical deformations of shells of revolution. The principal new results of the present analysis concern the form of strain-displacement and compatibility differential equations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):271-276. doi:10.1115/1.3564620.

The dynamic response of cylindrical and conical panels subjected to arbitrary time-varying load distributions is studied and appropriate equations are presented. Convenient trigonometric series are used, in conjunction with finite-difference methods, to reduce the governing equations to sets of matrix equations. The numerical solution procedure involves time integration, using an unconditionally stable implicit (Houbolt) scheme, together with a Gaussian elimination technique particularly suited to the banded matrices involved. Calculated results treat the effects of conicity and various support conditions on the structural response. Individual cases show quantitatively how cylindrical panel response is more sensitive than that of conical panels to changes in edge restraint.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):277-284. doi:10.1115/1.3564621.

When a circular cylindrical shell (plane strain) is subjected to a uniform radial impulse, the resulting circular mode may be unstable. In such a case flexural motion is excited, resulting in rather large displacements and stress. A previous nonlinear analysis [1]1 used a linear inextensionality constraint and displacement representation for the flexural response. A formulation employing a nonlinear inextensionality constraint is presented in this paper, and a comparison is made with the earlier work. The most significant result is a fundamental difference between the equations of motion; in this analysis the nonlinear modal coupling is primarily inertial. The condition for stability of the circular mode is unaffected, but substantial differences may occur in the long-term (nonlinear) response.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):285-291. doi:10.1115/1.3564622.

The von Karman equations for large deflections of an elastic circular plate containing a central hole and subjected to a concentrated ring load are presented in dimensionless and finite-difference form. Because of the nonlinear character of these equations an iterative technique must be employed to obtain a solution of the system of finite-difference equations and their corresponding boundary conditions. An analytical representation of the bounds within which the solution must lie is derived using a Green’s function approach. Finally, an example is solved numerically and the results discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):292-295. doi:10.1115/1.3564623.

The side-force problem for a shallow helicoidal shell is shown to be the complete static-geometric analog of the pure bending problem for the same shell. The solution of the former in terms of elementary functions is obtained, without another set of independent calculations, simply by applying the rules of the static-geometric duality to that of the latter. The analogy also enables us to use the same computer program developed for the pure bending problem (without any modification) to generate numerical results for the side-force problem.

Commentary by Dr. Valentin Fuster

DESIGN DATA AND METHODS

J. Appl. Mech. 1969;36(2):296-303. doi:10.1115/1.3564624.

Formulations are given for the coefficients λ, μ, ν defined by Hertz in terms of the solution of a transcendental equation involving elliptic integrals and used by him to describe the deformation of bodies subjected to contact stresses. Methods of approximate calculation are explained and errors in the tables prepared by Hertz are noted. For the purpose of providing a more extensive and more accurate tabulation, using an automatic digital computer, these coefficients are reformulated so that a large part of the variation is expressed by means of easily interpreted elementary formulas. The remainder of the variation is tabulated to 6 places for 100 values of the argument. Graphs of the coefficients are also provided.

Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Appl. Mech. 1969;36(2):304-307. doi:10.1115/1.3564625.

An attempt is made to indicate and explain some advantages of processes of cup drawing of metals which use a superposed fluid pressure. The mechanics of tensile necking failure is used to obtain a measure of relative drawability.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):307-309. doi:10.1115/1.3564626.
Abstract
Topics: Oscillations
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):309-311. doi:10.1115/1.3564627.

In this paper, the problem of incompressible laminar viscous flow in the annular space bounded by two coaxial infinite circular cylinders with an arbitrary time-varying pressure gradient and with an arbitrary initial distribution of velocity has been studied. The present problem generalizes the several earlier works in which the pressure gradient and the initial distribution of velocity have been taken in special forms. The analysis has been made by the use of finite Hankel transform. The case of steady flow when the pressure gradient is constant has been deduced by taking the pressure gradient to be a constant quantity and then letting the time since the start of the motion be infinite. This result has been shown in agreement with the already well-established result.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):311-313. doi:10.1115/1.3564628.

The stability of the equation Z̈ + f(t)Z = 0, where f(t) = (δ − ε cos2 t)/(1 − ε cos2 t), is studied by using Floquet theory, Fourier analysis and perturbations. The results are used to study the stability of the vibrations of a particle constrained to a plane and restrained by two identical linear springs with initial stress.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):313-316. doi:10.1115/1.3564629.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):316-318. doi:10.1115/1.3564630.

Typical web conveyance components are described by equivalent electrical circuits with web tension and velocity analogous to voltage and current, respectively. Such phenomena as waves and resonances in web handling system are easily understood in terms of the analogous electrical networks.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):318-320. doi:10.1115/1.3564631.
Abstract
Topics: Vibration , Shells
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):320-323. doi:10.1115/1.3564632.
Abstract
Topics: Strips
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):323-325. doi:10.1115/1.3564633.

Stresses and displacements in largely deflected cantilever beams subjected to gravity were analyzed by means of photoelasticity. The results obtained for stresses and moments are normalized and presented parametrically for increasing amounts of deflection. For comparison purposes the results obtained using elementary beam theory (small deflection) and large deflection theory are also included.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):325-327. doi:10.1115/1.3564634.

The values of the critical concentrated load, at which hinged-hinged circular arches begin to sway sideward from the symmetrical configuration at large deflections, are calculated by means of the exact theory of the inextensional elastica for initially circular rods of constant cross section. The calculations employ elliptic integrals of the first and second kind. The results are compared with available approximate and empirical values.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):330-333. doi:10.1115/1.3564637.
Abstract
Topics: Acoustics , Vibration
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):333-334. doi:10.1115/1.3564638.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):334-335. doi:10.1115/1.3564639.

A simple method is presented for measuring time-average velocity, turbulence intensity, and concentration of droplets in two-phase (air-liquid) turbulent flows.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):336-338. doi:10.1115/1.3564640.
Abstract
Topics: Boundary layers
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):338-340. doi:10.1115/1.3564641.

The reduction of the expected lifetime of a temperature-sensitive viscoelastic von Mises truss caused by small random temperature fluctuations is calculated, assuming the probability density of the temperature process to be a known (symmetric) function. Results are derived for a normal and a rectangular probability distribution of the temperature fluctuations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):340-342. doi:10.1115/1.3564642.
Abstract
Topics: Stress , Waves , Heating
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):342-345. doi:10.1115/1.3564643.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):345-346. doi:10.1115/1.3564644.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):347-348. doi:10.1115/1.3564645.

It is shown that to duplicate the combustion products of earth-storable liquid propellants by the combustion of a gaseous fuel and oxidizer combination which is at the same initial temperature, the only two conditions that must be satisfied are: (a) the elemental composition (i.e., atomic species and their proportions) of the liquid and gaseous propellants must be the same; (b) the initial energies (i.e., total enthalpies) of the two propellant combinations must be the same. An illustrative example of the simulation of the liquid propellant combination of N2 O4 /50 percent N2 H4 -50 percent UDMH at an O/F = 2 is given.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):350-352. doi:10.1115/1.3564648.
Abstract
Topics: Pipes , Buckling
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):352-355. doi:10.1115/1.3564649.

Two approximate solutions are given for Flügge’s closed cylindrical shell characteristic equation, valid for the lower and higher harmonics, respectively. It is shown that as the harmonic number increases, the roots of Donnell’s characteristic equation do not behave asymptotically like those of the Flügge equation.

Topics: Pipes , Equations
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):355-357. doi:10.1115/1.3564650.
Abstract
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):360-362. doi:10.1115/1.3564652.
Abstract
Topics: Oscillations
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):362-364. doi:10.1115/1.3564653.
Abstract
Topics: Pipes
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):364-365. doi:10.1115/1.3564654.
Abstract
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):370-372. doi:10.1115/1.3564657.

Equations for the absolute dimensions of the Karman vortex street are developed in terms of the coefficient of drag and the Strouhal number of the vortex shedding bluff body. The body is assumed to be of large slenderness ratio and of uniform cross section. The predicted vortex spacings are compared with the experimental results of other investigators for circular cylinders, flat plates, and a wedge.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):379-381. doi:10.1115/1.3564677.
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Abstract

BOOK REVIEWS

J. Appl. Mech. 1969;36(2):382. doi:10.1115/1.3564678.
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Abstract
Topics: Plasticity
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):382. doi:10.1115/1.3564680.
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Abstract
Topics: Thermodynamics
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):382. doi:10.1115/1.3564681.
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Abstract
Topics: Lubrication
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):382-383. doi:10.1115/1.3564682.
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Abstract
Topics: Acoustics
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):383. doi:10.1115/1.3564684.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):383. doi:10.1115/1.3564685.
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Abstract
Topics: Viscoelasticity
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):383. doi:10.1115/1.3564686.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1969;36(2):383-384. doi:10.1115/1.3564687.
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Abstract
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

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