THREE‐DIMENSIONAL GRAVITY INSTABILITY DERIVED FROM TWO‐DIMENSIONAL SOLUTIONS

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 153-166 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Periodic Patterns ◽  
Numerical Work ◽  
Characteristic Distance ◽  
General Validity ◽  
Dimensional Gravity ◽  
History Of

The theory of three‐dimensional gravity instability of multilayers is developed with particular application to salt structures. It is shown that three‐dimensional solutions are immediately obtained without further numerical work from the solution of the corresponding two‐dimensional problem. Application to a number of typical three‐dimensional structures yields the characteristic distance between peaks and crests and shows that this distance does not differ significantly from the wavelength of the two‐dimensional solution. Various periodic patterns are examined corresponding to rectangular and hexagonal cells. The time history of nonperiodic structures corresponding to initial deviations from perfect horizontality is also derived. The method is applied to the three‐dimensional problem of generation of salt structures when the time‐history of sedimentation is taken into account with variable thickness and compaction of the overburden and establishes the general validity of the geological conclusions derived from the previous two‐dimensional treatment of the same problem (Biot and Odé, 1965). The present method of deriving three‐dimensional solutions, which is developed here in the special context of gravity instability, is valid for a wide variety of problems in theoretical physics.

Time-dependent motions of the cavity formed when a uniform corpuscular flux is incident on the magnetic field of a line current

1968 ◽  
Vol 2 (1) ◽  
pp. 51-57 ◽  
Author(s):  
R. C. Hewson-Browne ◽  
D. N. Burghes
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Time History ◽  
Time Dependent ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Line Current ◽  
History Of ◽  
Steady State Solutions ◽  
The Magnetic Field

This paper concerns the time-dependent motions of the cavity formed when a uniform corpuscular flux is incident on the magnetic field of a line current. The two-dimensional problem is formulated and solved with two particular classes of solutions being given; namely, the steady-state solutions and the time-history of the interaction with a cloud of flux.

Mine Impact Burial Prediction From One to Three Dimensions

2008 ◽  
Vol 62 (1) ◽  
Author(s):  
Peter C. Chu
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Three Dimensions ◽  
Vertical Position ◽  
Burial Depth ◽  
Prediction Errors ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Models

The Navy’s mine impact burial prediction model creates a time history of a cylindrical or a noncylindrical mine as it falls through air, water, and sediment. The output of the model is the predicted mine trajectory in air and water columns, burial depth/orientation in sediment, as well as height, area, and volume protruding. Model inputs consist of parameters of environment, mine characteristics, and initial release. This paper reviews near three decades’ effort on model development from one to three dimensions: (1) one-dimensional models predict the vertical position of the mine’s center of mass (COM) with the assumption of constant falling angle, (2) two-dimensional models predict the COM position in the (x,z) plane and the rotation around the y-axis, and (3) three-dimensional models predict the COM position in the (x,y,z) space and the rotation around the x-, y-, and z-axes. These models are verified using the data collected from mine impact burial experiments. The one-dimensional model only solves one momentum equation (in the z-direction). It cannot predict the mine trajectory and burial depth well. The two-dimensional model restricts the mine motion in the (x,z) plane (which requires motionless for the environmental fluids) and uses incorrect drag coefficients and inaccurate sediment dynamics. The prediction errors are large in the mine trajectory and burial depth prediction (six to ten times larger than the observed depth in sand bottom of the Monterey Bay). The three-dimensional model predicts the trajectory and burial depth relatively well for cylindrical, near-cylindrical mines, and operational mines such as Manta and Rockan mines.

Effect of Finite Rotations on Gyroscopic Sensing Devices

1958 ◽  
Vol 25 (2) ◽  
pp. 210-213
Author(s):  
L. E. Goodman ◽  
A. R. Robinson
Keyword(s):  
Angular Velocity ◽  
Three Dimensional ◽  
Time History ◽  
Simple Method ◽  
Finite Rotations ◽  
Actual Solution ◽  
History Of ◽  
Body Components ◽  
Guidance Systems ◽  
Three Body

Abstract The well-known noncommutativity of three-dimensional finite rotations has long been a curiosity in mechanics since, in actual solution of dynamical problems, the angular velocity, which is conveniently representable as a vector, plays a more natural role. In modern inertial guidance systems, however, the orientation of a body in space, i.e., a rotation, is of primary engineering interest. In this paper a simple method of determining orientation from the time history of three body components of angular velocity is developed by means of a new theorem in kinematics. As a special case of this theorem it is shown that a gyro subjected to a regime of rotations which returns it to the original space orientation will, in general, produce a residual signal. It will have experienced a nonzero and easily calculated mean angular velocity about its input axis. Some implications of the theorem for the design of inertial guidance systems and for the testing of gyros are discussed.

Chiral modular bootstrap

2019 ◽  
Vol 34 (28) ◽  
pp. 1950168 ◽  
Author(s):  
M. Ashrafi
Keyword(s):  
Black Hole ◽  
Upper Bound ◽  
Three Dimensional ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Btz Black Hole ◽  
Conformal Field ◽  
Dimensional Gravity ◽  
Large Spin ◽  
Dimension Ratio

Using modular bootstrap we show the lightest primary fields of a unitary compact two-dimensional conformal field theory (with [Formula: see text], [Formula: see text]) has a conformal weight [Formula: see text]. This implies that the upper bound on the dimension of the lightest primary fields depends on their spin. In particular if the set of lightest primary fields includes extremal or near extremal states whose spin to dimension ratio [Formula: see text], the corresponding dimension is [Formula: see text]. From AdS/CFT correspondence, we obtain an upper bound on the spectrum of black hole in three-dimensional gravity. Our results show that if the first primary fields have large spin, the corresponding three-dimensional gravity has extremal or near extremal BTZ black hole.

scholarly journals Three-dimensional molar enamel distribution and thickness in Australopithecus and Paranthropus

2008 ◽  
Vol 4 (4) ◽  
pp. 406-410 ◽  
Author(s):  
A.J Olejniczak ◽  
T.M Smith ◽  
M.M Skinner ◽  
F.E Grine ◽  
R.N.M Feeney ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Cross Sectional ◽  
Enamel Thickness ◽  
Australopithecus Africanus ◽  
Interspecific Differences ◽  
Dimensional Measurements ◽  
Dimensional Distribution ◽  
History Of ◽  
Thickness Measurements

Thick molar enamel is among the few diagnostic characters of hominins which are measurable in fossil specimens. Despite a long history of study and characterization of Paranthropus molars as relatively ‘hyper-thick’, only a few tooth fragments and controlled planes of section (designed to be proxies of whole-crown thickness) have been measured. Here, we measure molar enamel thickness in Australopithecus africanus and Paranthropus robustus using accurate microtomographic methods, recording the whole-crown distribution of enamel. Both taxa have relatively thick enamel, but are thinner than previously characterized based on two-dimensional measurements. Three-dimensional measurements show that P. robustus enamel is not hyper-thick, and A. africanus enamel is relatively thinner than that of recent humans. Interspecific differences in the whole-crown distribution of enamel thickness influence cross-sectional measurements such that enamel thickness is exaggerated in two-dimensional sections of A. africanus and P. robustus molars. As such, two-dimensional enamel thickness measurements in australopiths are not reliable proxies for the three-dimensional data they are meant to represent. The three-dimensional distribution of enamel thickness shows different patterns among species, and is more useful for the interpretation of functional adaptations than single summary measures of enamel thickness.

Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor

1987 ◽  
Vol 54 (4) ◽  
pp. 772-777 ◽  
Author(s):  
R. M. Christensen
Keyword(s):  
Elastic Moduli ◽  
Fiber Composite ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Axial Deformation ◽  
Space Network ◽  
Plane Normal ◽  
Rank Tensor ◽  
Three Space

Symmetry conditions are found that assure isotropy of the fourth rank tensor of elastic moduli. Crystallography provides the answer to this problem in the two-dimensional context, namely one axis of three-fold symmetry assures the isotropy of properties in the plane normal to the axis. The present work provides the answer in the three-dimensional problem: 6 axes of five-fold symmetry are sufficient to give isotropy of the elastic moduli. An important restriction must accompany the present result. The derivation is given in the special form appropriate to low density materials which have a microstructure that transmits load according to the axial deformation of a space network of material distributed into micro-struts. The corresponding fiber composite idealization is that of a fiber dominated system, it therefore follows that if the fibers take the 6 specific orientations in three-space then isotropy is obtained.

A contribution to the application of photoelasticity to the micromechanics of composite materials

1969 ◽  
Vol 4 (2) ◽  
pp. 88-94 ◽  
Author(s):  
D E W Stone
Keyword(s):  
Composite Materials ◽  
Three Dimensional ◽  
Maximum Amount ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Casting Technique ◽  
Intensity Measurements ◽  
Photoelastic Model ◽  
Composite Models

Photoelastic-model methods can prove advantageous for the investigation of microstresses in composite materials. Some two-dimensional investigations of this type are discussed and the extension of this work into three dimensions is considered. It is suggested that more than one approach to the three-dimensional problem may be practicable, and special attention is paid to obtaining the maximum amount of information from a sandwiched polariscope by means of light-intensity measurements. A cold-casting technique for the fabrication of composite models is also described.

3D structure–2D plate interaction model

2019 ◽  
Vol 24 (10) ◽  
pp. 3354-3377 ◽  
Author(s):  
Matko Ljulj ◽  
Josip Tambača
Keyword(s):  
Elastic Layer ◽  
3D Structure ◽  
Three Dimensional ◽  
Interaction Model ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Dimensional Model ◽  
Limit Model ◽  
Two Dimensional Model ◽  
Thin Elastic Layer

In this paper, we derive models for the interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. Rigorous derivation is performed by considering a thin three-dimensional layer and the asymptotics of the solution of the full remaining three-dimensional problem when the thickness [Formula: see text] of the thin layer tends to zero. Furthermore, the attached thin material is assumed to have the elasticity coefficients which are of order [Formula: see text], for [Formula: see text] with respect to the coefficients of the three-dimensional body. In the limit, five different models are obtained with respect to different choices of p, namely [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Furthermore a three-dimensional–two-dimensional model is proposed that has the same asymptotics as the original three-dimensional problem. This is convenient for applications because one does not have to decide in advance which limit model to use.

Sign in / Sign up

Export Citation Format

Share Document