scholarly journals Isostasy with Love: I Elastic equilibrium

2021 ◽  
Author(s):  
Mikael Beuthe
Keyword(s):  
Boundary Conditions ◽  
Elastic Shell ◽  
Elastic Equilibrium ◽  
Gravity Anomalies ◽  
Analytical Form ◽  
The Body ◽  
Internal Stresses ◽  
Asymptotic State ◽  
Elastic Stresses ◽  
Love Numbers

Summary Isostasy explains why observed gravity anomalies are generally much weaker than what is expected from topography alone, and why planetary crusts can support high topography without breaking up. On Earth, it is used to subtract from gravity anomalies the contribution of nearly compensated surface topography. On icy moons and dwarf planets, it constrains the compensation depth which is identified with the thickness of the rigid layer above a soft layer or a global subsurface ocean. Classical isostasy, however, is not self-consistent, neglects internal stresses and geoid contributions to topographical support, and yields ambiguous predictions of geoid anomalies. Isostasy should instead be defined either by minimizing deviatoric elastic stresses within the silicate crust or icy shell, or by studying the dynamic response of the body in the long-time limit. In this paper, I implement the first option by formulating Airy isostatic equilibrium as the linear response of an elastic shell to a combination of surface and internal loads. Isostatic ratios are defined in terms of deviatoric Love numbers which quantify deviations with respect to a fluid state. The Love number approach separates the physics of isostasy from the technicalities of elastic-gravitational spherical deformations, and provides flexibility in the choice of the interior structure. Since elastic isostasy is invariant under a global rescaling of the shell shear modulus, it can be defined in the fluid shell limit, which is simpler and reveals the deep connection with the asymptotic state of dynamic isostasy. If the shell is homogeneous, minimum stress isostasy is dual to a variant of elastic isostasy called zero deflection isostasy, which is less physical but simpler to compute. Each isostatic model is combined with general boundary conditions applied at the surface and bottom of the shell, resulting in one-parameter isostatic families. At long wavelength, the thin shell limit is a good approximation, in which case the influence of boundary conditions disappears as all isostatic families members yield the same isostatic ratios. At short wavelength, topography is supported by shallow stresses so that Airy isostasy becomes similar to either pure top loading or pure bottom loading. The isostatic ratios of incompressible bodies with three homogeneous layers are given in analytical form in the text and in complementary software.

scholarly journals Elastoplastic bend of round steel beam. Message 2. Residual stresses

2018 ◽  
Vol 61 (11) ◽  
pp. 884-890
Author(s):  
V. N. Shinkin
Keyword(s):  
Residual Stresses ◽  
Extreme Values ◽  
General Concept ◽  
The Body ◽  
Internal Stresses ◽  
Metal Structures ◽  
Term Operation ◽  
Elastic Stresses ◽  
Deformation Hardening ◽  
Cylindrical Steel

The residual stresses in metals can lead to the defects in metals during their forming and to destruction of metal structures during their long-term operation. The resulting residual stresses during metal forming can be of plastic nature, as in the malleable metals, or caused by a slow irreversible creep at the increased temperatures and prolonged action of loads. In the viscoelastic mediums, it can be caused by the viscous parts of deformation that can accumulate when the body is deformed for a long period of time. The residual stresses also have an effect on the metals microstructure and can present inside and around the crystalline grains as the micro-residual stresses, which are called the hidden elastic stresses. Sometimes the residual stresses are called the eigenstresses by an analogy with the eigenfunctions, introduced by the mathematicians to denote the functions that correspond to the certain values (the eigenvalues) of parameters of the differential equation under the given boundary conditions. The concept of the internal stresses was proposed as a general concept for this type of stresses, created by the body itself; the term residual stresses is assigned to the case, when the internal stresses are caused by the irreversible deformation. In addition to the emergence of favorable system of residual stresses in the discs of malleable metals with a pronounced deformation hardening, there will also be a local increase in strength, provided that the Bauschinger’s effect does not negate the achieved advantages. The extreme values of residual stresses of a straight cylindrical steel rod (beam) during bending are studied below.

Boundaries Influence on the Flow Field Around an Oscillating Sphere

2013 ◽  
Author(s):  
Domenica Mirauda ◽  
Antonio Volpe Plantamura ◽  
Stefano Malavasi
Keyword(s):  
Free Surface ◽  
Flow Field ◽  
Boundary Conditions ◽  
Damping Ratio ◽  
Water Channel ◽  
Open Water ◽  
Free Surface Flows ◽  
The Body ◽  
Sphere Diameter ◽  
Oscillating Sphere

This work analyzes the effects of the interaction between an oscillating sphere and free surface flows through the reconstruction of the flow field around the body and the analysis of the displacements. The experiments were performed in an open water channel, where the sphere had three different boundary conditions in respect to the flow, defined as h* (the ratio between the distance of the sphere upper surface from the free surface and the sphere diameter). A quasi-symmetric condition at h* = 2, with the sphere equally distant from the free surface and the channel bottom, and two conditions of asymmetric bounded flow, one with the sphere located at a distance of 0.003m from the bottom at h* = 3.97 and the other with the sphere close to the free surface at h* = 0, were considered. The sphere was free to move in two directions, streamwise (x) and transverse to the flow (y), and was characterized by values of mass ratio, m* = 1.34 (ratio between the system mass and the displaced fluid mass), and damping ratio, ζ = 0.004. The comparison between the results of the analyzed boundary conditions has shown the strong influence of the free surface on the evolution of the vortex structures downstream the obstacle.

Numerical Solution of a Hydrofoil Moving Near an Interface

1996 ◽  
Vol 40 (04) ◽  
pp. 269-277
Author(s):  
G. X. Wu ◽  
T. Miloh ◽  
G. Zilman
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Body Surface ◽  
Iteration Scheme ◽  
The Body ◽  
Two Fluids ◽  
Linearized Theory

The problem of a hydrofoil moving near an interface of two fluids of different densities is analyzed. An iteration scheme is proposed which imposes the boundary conditions on the body surface and on the interface alternately. The numerical solution is obtained by using the linearized theory and a Glauert-type expansion for the vortex distribution. Results are provided for various cases with different densities and different speeds.

The Solution of Boundary Value Problems of Various Types with Consideration of Volume Forces for Anisotropic Bodies of Revolution

2021 ◽  
pp. 57-70
Author(s):  
D.A. Ivanychev ◽  
E.Yu. Levina
Keyword(s):  
Elastic Equilibrium ◽  
Transversely Isotropic ◽  
The Body ◽  
Bodies Of Revolution ◽  
State Spaces ◽  
Elastic Fields ◽  
Boundary State ◽  
Boundary States ◽  
Anisotropic Bodies ◽  
The Given

In this work, we studied the axisymmetric elastic equilibrium of transversely isotropic bodies of revolution, which are simultaneously under the influence of surface and volume forces. The construction of the stress-strain state is carried out by means of the boundary state method. The method is based on the concepts of internal and boundary states conjugated by an isomorphism. The bases of state spaces are formed, orthonormalized, and the desired state is expanded in a series of elements of the orthonormal basis. The Fourier coefficients, which are quadratures, are calculated. In this work, we propose a method for forming bases of spaces of internal and boundary states, assigning a scalar product and forming a system of equations that allows one to determine the elastic state of anisotropic bodies. The peculiarity of the solution is that the obtained stresses simultaneously satisfy the conditions both on the boundary of the body and inside the region (volume forces), and they are not a simple superposition of elastic fields. Methods are presented for solving the first and second main problems of mechanics, the contact problem without friction and the main mixed problem of the elasticity theory for transversely isotropic finite solids of revolution that are simultaneously under the influence of volume forces. The given forces are distributed axisymmetrically with respect to the geometric axis of rotation. The solution of the first main problem for a non-canonical body of revolution is given, an analysis of accuracy is carried out and a graphic illustration of the result is given

The Problem on Dynamical Loading of Plane Elastic Regions With Irregular Boundaries

1998 ◽  
Vol 65 (2) ◽  
pp. 476-478
Author(s):  
N. Morozov ◽  
I. Sourovtsova
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
The Body ◽  
Single Type ◽  
Edge Condition ◽  
Special Boundary ◽  
Elastic Wedge ◽  
Surface Loading ◽  
Special Surface ◽  
Extension Of Operators

The study of the problem of wave propagation in elastic wedge meets considerable difficulties, which are intensified by the presence of waves of two types that interact with each other through boundary conditions. However, some special surface loading permits separation of the potentials in the boundary conditions, but even in this case the problem cannot be simply reduced to two acoustic ones. The reason for this is that the edge condition cannot be satisfied if the disturbances are limited to a single type (longitudinal or shear). In spite of this the problem, such a special boundary loading nevertheless turns out to be very similar to the acoustic one, which makes it possible to find a closed analytical solution by means of the modified Kostrov method (Kostrov, 1966) and the idea of extension of operators. A similar approach is used for the study of the general problem of loading of the body with several angles.

Movement paths in operating hand-held tools: tests of distal-shift hypotheses

2013 ◽  
Vol 109 (11) ◽  
pp. 2680-2690 ◽  
Author(s):  
Sandra Sülzenbrück ◽  
Herbert Heuer
Keyword(s):  
Boundary Conditions ◽  
Visual Feedback ◽  
The Body ◽  
Hand Movements ◽  
First Order ◽  
Dynamic Transformation ◽  
Movement Characteristics ◽  
Effective Part

Extending the body with a tool could imply that characteristics of hand movements become characteristics of the movement of the effective part of the tool. Recent research suggests that such distal shifts are subject to boundary conditions. Here we propose the existence of three constraints: a strategy constraint, a constraint of movement characteristics, and a constraint of mode of control. We investigate their validity for the curvature of transverse movements aimed at a target while using a sliding first-order lever. Participants moved the tip of the effort arm of a real or virtual lever to control a cursor representing movements of the tip of the load arm of the lever on a monitor. With this tool, straight transverse hand movements are associated with concave curvature of the path of the tip of the tool. With terminal visual feedback and when targets were presented for the hand, hand paths were slightly concave in the absence of the dynamic transformation of the tool and slightly convex in its presence. When targets were presented for the tip of the lever, both the concave and convex curvatures of the hand paths became stronger. Finally, with continuous visual feedback of the tip of the lever, curvature of hand paths became convex and concave curvature of the paths of the tip of the lever was reduced. In addition, the effect of the dynamic transformation on curvature was attenuated. These findings support the notion that distal shifts are subject to at least the three proposed constraints.

Wooden Nano-Composite Materials and Prospects of their Application in Wooden Housing Construction

2018 ◽  
Vol 931 ◽  
pp. 583-588 ◽  
Author(s):  
Sergey I. Ovsyannikov ◽  
Vladislav Yurevich Dyachenko
Keyword(s):  
Wood Properties ◽  
The Body ◽  
Frame Structure ◽  
Seismic Resistance ◽  
Internal Stresses ◽  
Deep Processing ◽  
Nano Composite ◽  
Wood Structures ◽  
New Class ◽  
Natural Wood

Nano-composite material is a completely new class of material that combines wood pulp and some porous materials of artificial and natural origin. This is an artificially created material consisting of a polymer matrix of the porous natural or synthetic material. The number of micro or macro-pores in the composite can be different for different wood species variety of micro and macro capillaries varying in average from 25 to 35% of the wood volume. The change in wood properties occurs at the structuring of water-insoluble molecules smaller than 3 nm and that is a part of the filler. Industrial technology of deep processing of wood-based nanotechnology allows the manufacture of new products such as laminated wood structures with nano-device that have properties not existing in nature: 1. The wood becomes hydrophobic, it is characterised by almost complete lack of absorption by the body of the wood, which leads to almost full, the lack of swelling and the change of the geometrical sizes of the material; 2. The absence of cracking. As the penetrating substance is evenly distributed between micro and macro pores and uniformly fills all the frame structure, there are additional internal stresses, typical for products made of natural wood; 3. The use of such technology ensures high 10-25% of the density if you increase strength by 20%, which also increases the seismic resistance and mi CNTI products and structures.

A LEAST‐SQUARES PROCEDURE FOR GRAVITY INTERPRETATION

Geophysics ◽  
1965 ◽  
Vol 30 (2) ◽  
pp. 228-233 ◽  
Author(s):  
Charles E. Corbató
Keyword(s):  
Least Squares ◽  
High Speed ◽  
Gravity Anomalies ◽  
The Body ◽  
Two Dimensional ◽  
Partial Derivatives ◽  
Dimensional Gravity ◽  
Gravity Interpretation ◽  
Relationship Of ◽  
Derivatives Of

A procedure suitable for use on high‐speed digital computers is presented for interpreting two‐dimensional gravity anomalies. In order to determine the shape of a disturbing mass with known density contrast, an initial model is assumed and gravity anomalies are calculated and compared with observed values at n points, where n is greater than the number of unknown variables (e.g. depths) of the model. Adjustments are then made to the model by a least‐squares approximation which uses the partial derivatives of the anomalies so that the residuals are reduced to a minimum. In comparison with other iterative techniques, convergence is very rapid. A convenient method to use for both the calculation of the anomalies and the adjustments is the two‐dimensional method of Talwani, Worzel, and Landisman, (1959) in which the outline of the body is polygonized and the anomalies and the partial derivatives of the anomaly with respect to the depth of a vertex on the body can be expressed as functions of the coordinates of the vertex. Not only depths but under certain circumstances regional gravity values may be evaluated; however, the relationship of the disturbing body to the gravity information may impose certain limitations on the application of the procedure.

Hyperplastic Forming: Process Potential and Factors Affecting Formability

1999 ◽  
Vol 601 ◽  
Author(s):  
Glenn S. Daehn ◽  
Vincent J. Vohnout ◽  
Subrangshu Datta
Keyword(s):  
Boundary Conditions ◽  
Sample Size ◽  
Active Control ◽  
High Velocity ◽  
Size Dependence ◽  
Recent Model ◽  
The Body ◽  
Electromagnetic Forming ◽  
Forming Process ◽  
Factors Affecting

AbstractThis paper has two distinct goals. First, we argue in an extended introduction that high velocity forming, as can be implemented through electromagnetic forming, is a technology that should be developed. As a process used in conjunction with traditional stamping, it may offer dramatically improved formability, reduced wrinkling and active control of springback among other advantages. In the body of the paper we describe the important factors that lead to improved formability at high velocity. In particular, high sample velocity can inhibit neck growth. There is a sample size dependence where larger samples have better ductility than those of smaller dimensions. These aspects are at least partially described by the recent model of Freund and Shenoy. In addition to this, boundary conditions imposed by sample launch and die impact can have important effects on formability.

Assessment of Various Inviscid-Wall Boundary Conditions: Applications to NACA65 Compressor Blade

2011 ◽  
Author(s):  
Habib Khazaei ◽  
Ali Madadi ◽  
Mohammad Jafar Kermani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Solid Wall ◽  
Inviscid Flow ◽  
Compressor Blade ◽  
The Body ◽  
Wall Boundary Condition ◽  
Velocity Magnitude ◽  
Wall Boundary ◽  
Wall Boundary Conditions

Boundary condition is one of the major factors to influence the numerical stability and solution accuracy in numerical analysis. One of the most important physical boundary conditions in the flow field analysis is the wall boundary condition imposed on the body surfaces. To solve a three-dimensional compressible Euler equation (with five coupled PDE’s), totally five boundary conditions at the body surfaces should be prescribed. The momentum equation in the direction normal to the inviscid solid wall provides the pressure at the surface of the wall. For the cases with no-heat source or sink, the total temperature at the wall and the incoming flow should remain constant, when the steady condition is prevailed. The no-penetration condition through the solid wall and slip condition provides an equation relating the three velocity components. Assuming identical flow direction at the wall with the adjacent node, the last thing is the velocity magnitude that should be cast in such a way to give accurate, stable and robust solution. In this paper, four different methods for calculation of the wall velocity magnitude are proposed and applied to an identical test case of subsonic and supersonic flows such as: (1) Inviscid flow in a 3D converging-diverging nozzle, (2) Inviscid subsonic flow in a single 90° elbow, (3) Inviscid supersonic flow over a wedge, and (4) Inviscid flow through a compressor blade geometry of NACA 65410. A recently implemented 3D in-house CFD code (based on the flux difference splitting scheme of Roe (1981)) is used to compute compressible flows in generalized coordinates. It is found that the way to specify the additional numerical wall boundary condition strongly affects the overall stability and accuracy of the solution. It is concluded that there is no best boundary condition to cover all of the test cases, but the best wall boundary condition should be introduced very carefully for each type of flow.

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