The initial-value problem for elastodynamics of incompressible bodies

We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials. Our method consists in:

1. Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.
2. Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:

1. Using energy estimates to bound the growth of various Sobolev norms of solutions.
2. Finding the solution as the limit of a sequence of solutions of linearized problems.

Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.     

Inelastic deformations of mild-steel beams under symmetrical and unsymmetrical cyclic bending

The behavior of cantilever beams under the constraint of symmetrical and unsymmetrical cyclic deflections is investigated. An analytical technique developed predicts the behavior from cyclic moment-curvature relations derived from cyclic-strain control tests. Twenty tests were conducted on rectangular structural-steel sections under pure bending to establish these relations which couple the moment range and mean moment to curvature range. Models are proposed which fit the moment range and mean moment; these models are capable of accommodating, in discrete form, the phenomena of hardening and softening of a structural section as well as relaxation of the mean moment. Nine tests were conducted on cantilever beams under completely and partially reversed tip deflections. The load range changed little with changes in mean deflection. The mean load, in general, relaxed with increased cycling. The theory presented modelled the experimental behavior fairly accurately. It also suggested that the behavior may comprise:

1. an elastic case, where mean load is proportional to load range;
2. an intermediate range where the effects of mean deflection cannot be ignored; and
3. large inelastic cyclic deformation where the effects of mean deflection can be completely ignored except for associated changes in structural geometry and its secondary membrane effect.

     

Flow of a dipolar fluid between parallel plates

The flow of a dipolar fluid between two parallel plates with and without heat transfer is studied. The following cases are discussed:

1. Isothermal flow due to the relative motion of the plates,
2. Isothermal flow due to a constant pressure gradient with the plates at rest,
3. Nonisothermal flow with linearly varying plate temperatures.

Case (ii) is of particular interest to the experimentalists as it shows the effect of the material constants even when there are no externally applied dipolar tractions on the plates.     

The interaction of creep and fatigue for a rotor steel

Twenty tests were performed on a 1 Cr?1 Mo?1/4 V rotor steel at 1000° F (538°C) to determine the interaction of creep and low-cycle fatigue. These tests involved five different types of strain-controlled cycling: creep at constant tensile stress; linearly varying strain at different frequencies; and hold periods at maximum compressive strain, maximum tensile strain, or both. The experimental data were then used to characterize the interaction of creep and fatigue by the:

1. Frequency-modified strain-range approach of Coffin;
2. Total time to fracture vs. the time of one cycle relation as proposed by Conway and Berling;
3. Total time to fracture vs. the number of cycles to fracture characterization of Ellis and Esztergar;
4. Summation of damage fractions obtained from tests using interspersed creep and fatigue as proposed by the Metal Properties Council;
5. Strain-range-partitioning method of Manson, Halford, and Hirschberg.

In order to properly assess the strain-range-partitioning approach, seven additional tests were performed at the NASA Lewis Research Center. Visual, ultrasonic, and acoustic-emission methods of crackinitiation determination were unsuccessful. An approximate indication of crack initiation was obtained by finding the cycle N_o where the stress-cycle curve first deviated from a constant slope. Predictive methods (based on monotonic tests) for determining the fatigue life in the creep range were examined and found deficient, though they may still be useful for preliminary comparison of materials and temperatures. The extension of the frequency-modified strain-range approach to notched members was developed and the results of notched-bar tests were shown to corroborate this approach, when crack initiation for the plain and notched bars was campared.     

Oscillation modes of laval nozzle flow

Experimental investigations of Laval nozzle flow show for relatively low supply to exit pressure ratios, which correspond to shock wave positions close to the nozzle throat, three different, oscillatory instabilities.

1. Shock pattern oscillations where the root of a λ-like shock front remains nearly in constant position, but where the proportion between the normal part and the oblique part of the shock changes periodically.
2. Shock wave and separation bubble oscillations where the motion of the shock wave is accompanied by displacements of the separation bubble.
3. Flow rate oscillations where the shock waves leave periodically through the nozzle throat in upstream direction.

     

Die Berechnung des Blasenwachstums beim Verdampfen von Flüssigkeiten an Heizflächen als numerische Lösung der Erhaltungsgleichungen

A numerical procedure on the basis of the Marker and Cell-method [1] was developed in order to solve the conservation equations for mass, momentum and energy for the case of bubble growth on a heating surface. This procedure was used to calculate steam bubble growth on a horizontal stainless steel heating surface under saturated pool boiling conditions at a system pressure of 1 bar and different superheatings. The essential results obtained are:

-Good agreement was found between calculations and experiments concerning bubble growth rates, bubble shape and temperature field in the liquid surrounding the bubble.

-During its growth the bubble penetrates the temperature boundary layer formed in the liquid on the heating surface, simultaneously liquid is displaced aside.

-The microlayer evaporation fraction of the total bubble growth increases with growth time from 20 % to 50%.

     

Some pending problems in polymer melt rheology,as seen from the point of view of Doi's slip-link model

An exposition is given of results, as obtained with the aid of Doi's sliplink model, being considered as the most simple version of the famous “reptation model”. It turns out that this model which exhibits three distinct phases of relaxation (an extremely fast phase, an equilibration phase and a slow disengagement phase) is capable of explaining several peculiar features of polymer melt rheology:

1. The molecular mass dependence of the breadth of the rubber plateau in the storage modulus, of the zero-shear viscosity and of the normal-stress coefficients.
2. The molecular mass independence of the equilibrium (shear and tensile) compliances for monodisperse polymers (semi-quantitative prediction).
3. The seemingly contradictory sensitivity of these compliances for the breadth of the molecular mass distribution.
4. The critical value of the shear stress at which melt fracture occurs in capillary flow.
5. An equilibration phase in tensile experiments on unvolcanized rubber.

In this evaluation optical (flow birefringence) measurements are preferentially used.     

Variable isochromatic/isopachic-fringe visibility in photoholoelasticity

This paper presents a new “hybrid” method whereby the ratio of the isochromatic-fringe visibility/isopachic-fringe visibility may be easily and continuously varied. This simple procedure merely combines a conventional polariscope with a holographic system. A variable beam splitter permits an incoherent superposition of the reconstruction of a doubly exposed hologram with real-time isochromatics, either dark or light field. By varying the ratios of the above two, in the image plane, numerous interesting results may be obtained including:

1. Isochromatics only, without errors in position
2. Isochromatics-isopachic fringes identical to those obtained through classical interferometry
3. Isochromatic-isopachic fringes whereby the amplitude modulation between the two may be minimized
4. Continuously variable isopachic/isochromatic-fringe visibility.

     

Experimental determination of viscosity of rocks

An important parameter involved in the viscoelastic deformation of structural materials is the coefficient of “solid” viscosity. Determination of this parameter is necessary, if it is to be used in structural design. This paper deals with pertinent analytical considerations concerning solid viscosity and describes the procedures followed in the determination of parameters for structural and true viscosity of a Queenston limestone. The following three techniques were used:

1. Relaxation technique
2. Uniaxial compressive loading
3. Cantilever-beam loading

The results obtained are in close mutual agreement except for (2) above, where experimental conditions were different from those in (1) and (3). A quasi-periodic behavior of strain is indicated. It has been shown that the solid viscous parameter is a transient property and may depend on such factors as applied load, time, grain size, grain-packing in a material, and the direction of testing. It has been concluded that coefficients of true and structural solid viscosity of materials can be determined for a given set of conditions.     

A fast and accurate graphical method for designing an air conditioning cooling coil

The aim of this paper is to develop a fast and simple accurate graphical method for designing the required cooling coil for an air conditioning system in which both the sensible and latent heat are transferred. The method has the following advantages:

1. Direct solution utilizing only Psychrometer charts.
2. Solution of combined heat and mass transfer problems occuring at pressures other than that of atmosphere.
3. Direct determination of boundary temperature at which dehumidification begins for that type of problem where a portion of the surface is in a dry condition.
4. Less effort required than trial and error method in determination of air condition leaving a counterflow coil of a given area.

The calculated values of the method showed a good agreement with the experimental results. The average deviation for the total heat is about + 15 % and for sensible heat is + 9 %, which are of positive nature and on the safe side for practical design purposes. The method is also useful for similar practical application.     

Glass-bonding techniques for semiconductor strain gages

Conventional organic-epoxy adhesives outgas when exposed to ultra-high vacuum and, as operating temperatures are increased, they begin to exhibit plastic behavior causing hysteresis and zero instability in the transducer. The use of an inorganic glass as the bonding material has resulted in a significant advance in transducer-fabrication technology for the following reasons:

1. The outgassing of transducers in high-vacuum applications is minimized.
2. Mechanical properties of the transducer such as hysteresis and repeatability are improved.
3. The electrical isolation of the strain gages from the metallic elements of the transducer is increased at high temperatures over that provided by epoxy. Also, the glass bond can survive and operate in severe radiation environments, wherein the epoxy adhesive will suffer either temporary or permanent loss of its dielectric strength.
4. Glass-bonding techniques are particularly useful for the extension of the temperature range of operation of silicon-strain-gage transducers.

Nispan C and 440-C stainless-steel substrates were successfully used with glass-bonded silicon strain gages to fabricate transducers for evaluation.     

On the steady motion of two immiscible fluids in an undulating porous reservoir

We describe steady two-dimensional flows of two immiscible fluids through an undulating porous medium of constant thickness, with impermeable or slightly permeable boundaries. Flows in the same or opposite directions are called, respectively, direct or counter flows. Three special classes of flow are determined:

1. The pressure dominated case occurs for high direct flows and has the interface approximately a constant vertical distance from the impermeable boundaries.
2. The gravity dominated case occurs for low direct flows and has the interface very close to the lower (upper) boundary for downward (upward) sloping boundaries except at crossovers.
3. Counter flows require the interface to decrease in the direction of flow of the lower fluid.

Numerical examples illustrate the three classifications above. For incompressible flows the interface and pressure equations uncouple. A stability analysis shows that the direction of integration of the differential equation for the interface must be opposite to the flow direction for direct flows; for counter flows the direction of integration depends on whether the interface is above or below a critical height. Direct flows through cyclic geometries are asymptotically cyclic upstream. If the reservoir is ‘leaky’, asymptotically self-similar flows result when the (small) permeability ratio is scaled to the dynamical flow parameters.     

Rapid interpretation of moiré photographs

The paper describes the possibilities of rapidly and conveniently obtaining results by means of the reflection moiré method. The moiré photographs are produced with the aid of a screen provided with a finely ruled grid. The contrast of the photographs is increased by optical means. By displacement of two equal moiré photographs, moiré fringes of second order for the curvature or the twist are obtained. It is shown how charts can be produced which indicate:

1. The distribution of the bending moments
2. The distribution of the bending moments for which the reinforcement should be designed
3. The distribution of the shear forces and the magnitude of the bearing reactions.

     

Marangoni convection at alcohol aqueous solutions-air interfaces

For aqueousn-heptanol solutions and in a nearly two-dimensional flow, two strikingfeatures have been detected:

1. a shift of the minimum of the surface tension
2. a discrepancy between the observed Marangoni flow velocities and the expected ones from static surface tension values.

A qualitative explanation is given.     

Effects of artificial cracks and poisson's ratio upon photoelastic stress-intensity determination

A series of stress-freezing photoelastic experiments were performed with multiple replications upon edge-cracked strips for three types of “cracks” in current use:

1. Rectangular slots 0.152 mm wide,
2. 1.59-mm-wide slots terminating in a 30-deg vee notch of approximately 0.025-mm root radius, and
3. Natural cracks (approximately 0.0025-mm root radius).

Stress-intensity results were compared with the Gross-Srawley analysis; in addition (1) was compared with Savin's solution. It was concluded that (2) and (3) yield the same results but (1) was slightly higher. Both (2) and (3) were about 12 percent higher than the Gross-Srawley results. This is shown to be related to a Poisson's ratio effect.     

Interferogram processing by spatial filtering

This paper describes a technique of interferogram processing by the use of spatial-frequency filtering. Proper part of Fourier spectrum of the interferogram is blocked out by a matched mask located in the focal plane of the imaging system. Depending on the mask shape and dimensions one can obtain

1. interferogram images with delineating fringes or
2. interferogram images of one component of strain.

     

A logrithmic bound on the location of the poles of the scattering matrix

It is shown that the complex poles z of the scattering matrix satisfy the inequality: Im z≧a+b log ¦z¦, b>0, in three instances of classical scattering in three space dimensions described by the wave equation u_{t t}?c²Δu+qu=0.

1. c and q smooth with c=1 and q=0 for ¦x¦>p, all rays going to infinity, and the energy form positive definite.
2. c=1 and q=0 outside of a convex body on which u=0.
3. c=1, q bounded and measurable, q=0 for ¦x¦>p, and the energy form not necessarily positive definite.

     

Stress concentrations around horizontal circular tunnels

The problem of axial variation of stress concentrations at the periphery and normal to the axis of a circular tunnel is solved by means of the three-dimensional photoelasticity technique, under the following conditions:

1. The center lines of two horizontal tunnels of equal diameter (2r) are separated by a distanceK and include an angle α.
2. K and α assume values of 0, 3r, 7/2r, 4r and 30 deg, 60 deg, 90 deg, respectively.
3. The tunnels are located in a uniform, uniaxial stress field normal to the axes of the tunnels.

     

Flow past a suddenly cooled horizontal plate

The problem of unsteady laminar, incompressible free convection above a horizontal semi-infinite flat plate is studied theoretically. It is assumed that for timet<0 the plate is hotter than its surroundings and at timet=0 the plate is suddenly cooled to the same temperature of its surroundings. Three solutions of the momentum and energy equations are obtained, namely

1. an analytical solution which is valid for small time,
2. an asymptotic analytical solution which is valid for large time, and
3. a numerical solution which matches these two limiting analytical solutions.

It is found that the numerical solution matches the small and large time solutions accurately. Finally, the variation of the velocity, temperature, skin friction and heat transfer on the plate with time are discussed.     

Some characteristics of supersonic flow of an electrically conductive gas in an MHD-channel

We study supersonic flows of an electrically conductive gas in crossed electric and magnetic fields [1] in the presence of shock waves. It is shown that three steady flow regimes can exist, and that these are defined by the electrical conductivity of the gas as a function of temperature and density.

1. The normal regime is characterized by a tendency for the shock to move toward the channel entrance on increase of the static pressure at the channel exit. The steady regime of this type exists and is stable.
2. The anomalous regime (formally constructed) is characterized by a tendency for the shock to move toward the exit on increase of the static pressure at the channel exit. This regime is unstable and the flow in the MHD-channel may be either entirely supersonic or entirely subsonic.
3. The limiting (boundary) regime is intermediate between the normal and anomalous regimes and is characterized by the fact that the stationary position of the shock wave and its amplitude are not uniquely defined. Steady flow in this case is not unique.

This study involves formal construction both of the solution to the steady-state problem and the corresponding nonsteady-state problem [4]. The establishment of a steady regime in the solution of the unsteady problem, is at the same time, a verification of its stability.