Displacement field around a circular hole in a viscoelastic plate

Moiré experimental techniques are used to measure displacement fields in viscoelastic plates undergoing large deformations at elevated temperatures. These experimental procedures are applicable to determining displacement fields in nonlinear materials. As preliminary information, the material properties are determined from creep studies. The moiré method is used to determine the strains under constant load and isothermal conditions. Tests are conducted for several combinations of load and temperature for 2.5 decades of time. Assuming thermorheologically simple behavior, the data are shifted to establish the creep extensional compliance over ten decades in time. The constitutive equations are formulated as integral equations, the kernels of which are the functions that were measured in this work. These equations are solved exactly for the infinitesimal case. The finite case is then approximated by an incremental superposition of a series of successiye infinitesimal solutions. The results are applied to a plate initially containing a circular hole, and are shown to agree closely with the experimental measurements.     

Note on the reciprocity theorem for electrical systems

Summary It is shown by means of Maxwell's equations that the admittance matrix of a passive linear multipole placed in a constant magnetic fieldB ₀ satisfies the extended reciprocity relationG _ik(B ₀)=G _ki(–B ₀) if the conductivity, permittivity and permeability tensors of the materials forming the multipole all satisfy the spatial symmetry relationT (B ₀)=T (–B ₀).     

Axial diffusion in gas blanketed vapor condensers

A model is given, which permits determination of the height of a zone of mixing between a gas and a vapor in a condenser. The resulting predictive equation is algebraically simple and contains a non-dimensional group, z_i ^*, which should characterize a host of related problems: $$z_i^* = z_i \sqrt {\frac{{4U\Delta T_0 \tilde RT_0 }}{{pD_0 R\tilde h_{fg} }}} .$$ Data are presented which verify the mathematical model.     

On the influence of a flexural biasing state on the velocity of piezoelectric surface waves

A system of linear electroelastic equations for small fields superposed on a bias is applied in the determination of the velocity of acoustic surface waves in piezoelectric substrates subject to flexural biasing stresses. The influence of the biasing stresses appears in the boundary conditions as well as the differential equations. Direct calculations performed for both quartz and lithium niobate when the spatial variation of the flexural biasing state is omitted indicate that the biasing stresses in the boundary conditions have an important influence of the surface wave velocity. In addition, perturbation calculations are performed which include the influence of the spatial variation of all flexural biasing terms and it is shown that, for substrate thickness-to-wavelength ratios well within the practical range, the spatial variation in the biasing state has an appreciable effect on the velocity of acoustic surface waves.     

Das Digitale Beukenmodell mit nichtkonstanten Koeffizienten — eine Matrixmethode zur Lösung nichtlinearer Ausgleichsvorgänge

Zusammenfassung Mit Hilfe des Digitalen Beukenmodells mit nichtkonstanten Koeffizienten lassen sich ein- und mehrdimensionale Ausgleichsvorgänge berechnen. Besonders einfach ist das Verfahren, wenn nur die Wärmeleitfähigkeit von der Temperatur oder der Diffusionskoeffizient von der Konzentration abhängig ist. Der Hauptvorteil des Digitalen Beukenmodells liegt in der einfachen Handhabung. Für viele praktische Fälle kann die Rechenzeit gegenüber dem herkömmlichen reinen Differenzenverfahren entscheidend verkürzt werden. Das Digitale Beukenmodell mit nichtkonstanten Koeffizienten ist die genaueste Rechenmethode für nichtlineare Ausgleichsvorgänge.

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The digital Beuken-Model with non-constant coefficients — a matrix method for the solution of non-linear diffusion processes

By the help of the digital Beuken-Model with nonconstant coefficients it is possible to compute one- or multidimensional diffusion processes. Especially simple is that method, if only the thermal conductivity depends on the temperature or if the diffusion coefficient depends on the concentration. The main advantage of the digital Beuken-Model is its simple handling. In many practical cases the computing time can be shortened decisively to the traditional difference methods. The digital Beuken-Model with non-constant coefficients is the most exact computing-method for the solution of nonlinear diffusion processes.

     

Determination of the critical loads of shells by nondestructive methods

Two methods for determining the location of and load level to produce instability of compressed cylindrical shells are presented. The first relates the variation in the wall normal stiffness as a function of applied compressive force to the critical load. It uses the distribution of stiffness over the surface of the shell as a guide to buckle location. The second method associates the local dynamic mass with instability behavior. The test data presented show that either method will give excellent prediction capability from low-load-level data for shells of orthodox form. Neither method appears to apply to spirally stiffened shells. This is thought to be due to the fact that there is a substantial difference between the buckle pattern under axial compression and the imperfection shape induced by the normal displacement which is used to ascertain the wall stiffness and the dynamic mass.     

Experimental study of multiple interior impacts

In this paper, experiments involving interior impacts on soft, lead targets are described. The term “interior impact” refers to the impact of a projectile at the bottom of a predrilled hole in the target. It is known that impact of low-velocity projectiles into such soft targets results in plastic flow and cavities similar to those obtained in high-velocity impact of hard targets. Therefore, comparatively simple rifle-propelled projectile impacts on soft targets may yield useful information for high-speed hard-target impacts. Double impacts with a short-time interval, on the order of microseconds, between projectile arrivals were conducted. The leading projectile creates a plastic flow in the target which partially seals the entrance hole. The trailing projectile must first force open the passage hole before achieving additional penetration. The total penetration was measured for different time intervals between the projectiles. A method was developed to obtain double impacts at the same point. The method utilizes a duplex round, which is made up of two separate projectiles fired from the same cartridge. With this round, a series of impact experiments was conducted. It was learned that occlusion, or the close up of the hole, is negligible for nearly simultaneous projectiles (less than 10 μs between impacts), and greatest with about 80 to 100 μs between impacts. The projectile velocity was measured by the use of properly placed photodiodes.     

On conservative elastic systems of the first and second kind

Summary In introducing the definition of a generalized kind of selfadjointness, it can be shown that there are truly nonconservative systems in the classical sense which in the light of a generalized concept are, however, conservative systems of the second kind. Systems like these buckle by divergence, do possess a generalized conservation theorem, and a generalized Rayleigh's quotient. Examples of such systems are given: the pinnedpinned rod, the rod with sliding ends, and the simply supported plate, all of them subjected to compressive follower forces. An example involving a complex differential equation is also treated: Greenhill's problem of the torsion of a pinned-pinned shaft.

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Übersicht Durch Einführen eines verallgemeinerten Begriffs der Selbstadjungiertheit kann gezeigt werden, daß Systeme, die im klassischen Sinne nichtkonservativ sind, bei Zugrundelegen des neuen Begriffs als konservative Systeme zweiter Art aufgefaßt werden können. Für Systeme dieser Art gilt ein verallgemeinerter Erhaltungssatz und es existiert ein verallgemeinerter Rayleigh-Quotient. Einige Beispiele für derartige Systeme werden betrachtet: der beidseitig gelenkig gelagerte Balken, der Balken mit gleitenden Enden und die einfach gelagerte Platte — sofern die Belastung in allen drei Fällen durch mitgehende Druckkräfte geschieht. Es wird außerdem das Greenhillsche Problem des gelenkig gelagerten Stabes mit Torsion untersucht, wobei eine komplexe Differentialgleichung erhalten wird.

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The support of the National Research Council of Canada under Grant A7297 is gratefully acknowledged.     

Two new optical techniques to measure strain

A grating engraved on the surface to be analyzed is utilized to measure strain. The grating is illuminated by a monochromatic, plane wavefront. A lens produces the diffraction spectrum of the grating, and an opaque screen is located in the plane of the spectrum. A window allows a single diffraction order to go through the screen. Two techniques are presented. In the point-by-point technique, the illuminating wavefront is reduced to a very small area. Behind the window a light sensor detects the changes of light intensity that are produced by loading the specimen. In the field technique, an image of the specimen is produced, and the light sensor is located in the image plane. In both cases, changes of light intensity are related to strains. Using crossed gratings, a rosette strain gage is obtained. Examples of application of both techniques are presented, showing a very good agreement between the strains measured by the proposed methods and by independent means.     

The impact of high order refraction on optical microbubble sizing in multiphase flows

High order refractions have been found to have a great impact on optical particle sizing using phase-Doppler anemometry (PDA) especially when the relative refractive index of the media is less than one. The high order refractions cannot be neglected because of the differences in spatial frequencies and motion directions of each order. A model of phase-size correlation is proposed that considers these high order refractions. By using the conversion factors of the high order refractions, the particle diameter can be determined. The capability of the newly developed method was evaluated by using generalized Lorenz Mie theory (GLMT), which was validated by experiments. The results of the simulation and experiments are compared with those based on the conventional method. An optimization method for accurately sizing air bubbles in water has been suggested.