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.

     

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.     

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.

     

The approach of solutions of nonlinear diffusion equations to travelling front solutions

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation u_t—u_xx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.

1. Let u(x, 0)=u ₀(x) satisfy 0≦u ₀≦1. Let \(a\_ = \mathop {\lim \sup u0}\limits_{x \to - \infty } {\text{(}}x{\text{), }}\mathop {\lim \inf u0}\limits_{x \to \infty } {\text{(}}x{\text{)}}\) . Then u approaches a translate of U uniformly in x and exponentially in time, if a_? is not too far from 0, and a₊ not too far from 1.
2. Suppose \(\int\limits_{\text{0}}^{\text{1}} {f{\text{(}}u{\text{)}}du} > {\text{0}}\) . If a _? and a ₊ are not too far from 0, but u₀ exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts.
3. Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.

     

Fracture data and stress-strain behavior of rocks in triaxial compression

Test results are reported for a recently completed experimental research program on rocks subjected to triaxial compression. Sandstone, marble, granite and shale specimens were tested at confining pressures as high as 90,000 psi corresponding to mean stresses of up to 143,000 psi. Recognizing that the largest potential experimental error in such tests results from making strain and load measurements external to the vessel, special load and strain-measuring devices were designed and fabricated for use inside the pressure vessel. The specimens were carefully machined cylinders with length-to-diameter ratios of two and with diameters ranging from 4/16 in. to 1 in. The confining pressure was held constant during each run, but varied from 0 to 90,000 psi over the tests. Results are reported in the form of:

1. Stress-strain curves for individual specimens
2. Maximum shear stress at fracture vs. mean-stress curves for each rock type tested
3. Tabulation of results for 59 specimens

A number of tests were run on granite specimens which had been previously fractured. Results from these tests showed good agreement with tests on intact granite, providing the confining pressure was above 30,000 psi.     

Dynamics as a mechanism preventing the formation of finer and finer microstructure

We study the dynamics of pattern formation in the one-dimensional partial differential equation $$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$ proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u _x )² ?1)². What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: $$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$ Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]:

?While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (u _nv_n) of E[u,v] in the Sobolev space W ^{1 p}(0, 1)×L²(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W ₁ ^p(0,1)×L²(0,1) of all solutions (u(t),u_t(t)) with low initial energy as time t → ∞).

Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data — those where the strain field u _x has a transition layer structure; our analysis includes proofs that

?at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

?transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time t → ∞

?the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.

     

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.     

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.

     

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.     

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y?= ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In K _G and variance ${\sigma_{Y}^{2}}$ ) and of axisymmetric autocorrelation of integral scales I _h,I _v (anisotropy ratio f?=?I _v/I _h?<?1). The head gradient and the velocity are uniform in the mean, parallel to bedding, and of constant and given as J and U, respectively. Transport is ruled by advection, which typically overwhelms pore scale dispersion in the breakthrough curve (BTC) determination. In the present study we analyze the impact of anisotropy f on the BTC of a passive solute, which is related to the mass flux??? (t, x) at a control plane at x. While a considerable body of literature dealt with weakly heterogeneous formations ( ${\sigma _{Y}^{2} <1 }$ ), the present study addresses the case of ${\sigma _{Y}^{2} >1 }$ , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity K _ef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e.,??? (t, x) I _h/U depends only on ${\sigma _{Y}^{2}}$ (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.     

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.     

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.

     

Guidance properties of nonlinear planar waveguides

We discuss the propagation of electromagnetic waves through a stratified dielectric. The ability of such a device to support guided waves depends upon the way in which the refractive index varies across the layers. In the present discussion, we show how nonlinear effects and appropriate stratification can be used to obtain any one of the following behaviours:

1. guidance occurs only at low power.
2. guidance occurs only at high power.
3. guidance occurs at all powers.
4. there is no guidance.

The situation (i) is obtained by using materials with a defocusing dielectric response, whereas the situation (ii) is obtained for suitable configurations of self-focusing materials. The situations (iii) and (iv) can be obtained by using either defocusing or self-focusing materials. By seeking solutions of a particular form, we reduce the problem to the study of solutions in the Sobolev space H ¹(?) of a second-order differential equation. The discussion of defocusing nonlinearities is based in the study of the global behaviour of the branch of solutions bifurcating from a simple eigenvalue. For self-focusing nonlinearities we use a variational approach.     

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.     

Pressure drop in alternating curved tubes

Pressure drop measurements in the laminar and turbulent regions for water flowing through an alternating curved circular tube (x=h sin 2πz/λ) are presented. Using the minimum radius of curvature of this curved tube in place of that of the toroidally curved one in calculating the Dean number (ND=Re(D/2R _c )², it is found that the resulting Dean number can help in characterizing this flow. Also, the ratio between the height and length of the tube waves which represents the degree of waveness affects significantly the pressure drop and the transition Dean number. The following correlations have been found:

1. For laminar flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.03,\operatorname{Re}< 2000.$$
2. For turbulent flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.005,2000< \operatorname{Re}< 15000.$$
3. The transition Dean number: $$ND_{crit} = 5.012 \times 10^3 \left( {\frac{D}{{2R}}} \right)^{2.1} ,0.0111< {D \mathord{\left/ {\vphantom {D {2R_c }}} \right. \kern-\nulldelimiterspace} {2R_c }}< 0.71.$$

     

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.     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

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.     

Investigation of a vibrating piezoelectric ceramic disk by synthesis of optical coherent methods

Modes of a vibrating disk, made of piezoelectric ceramic-type PZT-5A, were investigated by two optical coherent methods:

1. Time-average holography (TAH).
2. Speckle-shearing interferometry (SSI). It is shown how synthesis of data from the two methods can give a good picture of the modes of vibration, more exact than the picture one can derive from each method alone.

     

Further studies on dynamic crack branching

The newly derived dynamic-crack-branching criterion with its modifications is verified by the dynamicphotoelastic results of dynamic crack branchings in thinpolycarbonate, single-edged crack-tension specimens. Successful crack branching was observed in four specimens and unsuccessful branching in another. Crack branching consistently occurred when the necessary conditions ofK _I =K _{I b} =3.3 MPa \(\sqrt m\) and the sufficiency condition ofr _o =r _c =0.75 mm were satisfied simultaneously. In the unsuccessful branching test, the necessary condition was not satisfied sinceK _I was always less thanK _{I b} .