scholarly journals Static Mantle Density Distribution 2 Improved Equation and Solution

2020 ◽  
Vol 3 (1) ◽  
Keyword(s):  
Potential Energy ◽  
Density Distribution ◽  
Integral Equations ◽  
Centrifugal Force ◽  
Minimum Potential Energy ◽  
Positive Mass ◽  
Negative Mass ◽  
Fredholm Type ◽  
Line Angle ◽  
Minimum Potential

Using Archimedes Principle of Sink or Buoyancy (APSB), Newton’s universal gravity, buoyancy, lateral buoyancy, centrifugal force and Principle of Minimum Potential Energy (PMPE), this paper improves the derivation of equation of static mantle density distribution. It is a set of mutual related 2-D integral equations of Volterra/Fredholm type. Using method of matchtrying, we find the solution. Some new results are: (1) The mantle is divided into sink zone, neural zone and buoyed zone. The sink zone is located in a region with boundaries of an inclined line, angle α_0=35°15’, with apex at O(0,0) revolving around the z-axis, inside the crust involving the equator. Where no negative mass exists, while positive mass is uniformly distributed. The buoyed zone is located in the remainder part, inside the crust involving poles. Where no positive mass exists, while negative mass is uniformly distributed. The neural zone is the boundary between the buoyed and sink zones. The shape of core (in sink zone) is not a sphere. (2) The total positive mass is equal to the total negative mass. (3) The volume of BUO zone is near twice the volume of SIN zone. (4) No positive mass exists in an imagine tunnel passing through poles (the z-axis).

scholarly journals Static Mantle Density Distribution 1 Equation

2019 ◽  
pp. 1-8 ◽  
Author(s):  
Tian Quan Yun
Keyword(s):  
Potential Energy ◽  
Density Distribution ◽  
Centrifugal Force ◽  
Seismic Waves ◽  
Vertical Direction ◽  
Gravitational Acceleration ◽  
Minimum Potential Energy ◽  
Fredholm Type ◽  
Minimum Potential ◽  
Two Parameters

The study of mantle distribution does relate to the reflecting of seismic waves, and has important meaning. Using Archimedes Principle of Sink or Buoyancy (APSB), Newton’s gravitation, buoyancy, lateral buoyancy, centrifugal force and the Principle of Minimum Potential Energy (PMPE), we derive equation of static mantle density distribution. It is a set of double-integral equations of Volterra/Fredholm type.  Some new results are: (1) The mantle is divorced into sink zone, neural zone and buoyed zone. The sink zone is located in a region with boundaries of a inclined line, with angle α1=35°15’ apex at  0(0,0,0) revolving around the z-axis, inside the crust involving the equator. The buoyed zone is located in the remainder part, inside the crust involving poles. The neural zone is the boundary between the buoyed and sink zones. The shape of core (in sink zone) is not a sphere. (2) The Potential energy inside the Earth is calculated by Newton’s gravity, buoyancy, centrifugal force and lateral buoyancy. (3) The gravitational acceleration above/on the crust is tested by formula with two parameters reflecting gravity and centrifugal force, and the phenomenon of “heavier substance sinks down in vertical direction due to attraction force, and moves towards to edges in horizontal direction due to centrifugal force” is tested by a cup of stirring coffee.

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Optimization for Minimum Potential Energy, Maximum Rigidity and Minimum Deformability

1984 ◽  
pp. 9-87
Author(s):  
Andrej M. Brandt ◽  
Wojciech Dzieniszewski ◽  
Stefan Jendo ◽  
Wojciech Marks ◽  
Stefan Owczarek ◽  
...  
Keyword(s):  
Potential Energy ◽  
Minimum Potential Energy ◽  
Minimum Potential ◽  
Energy Maximum

A Global Extremum Principle in Mixed Form for Equilibrium Analysis With Elastic/Stiffening Materials (a Generalized Minimum Potential Energy Principle)

1994 ◽  
Vol 61 (4) ◽  
pp. 914-918 ◽  
Author(s):  
J. E. Taylor
Keyword(s):  
Potential Energy ◽  
Problem Formulation ◽  
Extremum Problem ◽  
Equilibrium Analysis ◽  
Minimum Potential Energy ◽  
Problem Statement ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Minimum Potential ◽  
Minimum Potential Energy Principle

An extremum problem formulation is presented for the equilibrium mechanics of continuum systems made of a generalized form of elastic/stiffening material. Properties of the material are represented via a series composition of elastic/locking constituents. This construction provides a means to incorporate a general model for nonlinear composites of stiffening type into a convex problem statement for the global equilibrium analysis. The problem statement is expressed in mixed “stress and deformation” form. Narrower statements such as the classical minimum potential energy principle, and the earlier (Prager) model for elastic/locking material are imbedded within the general formulation. An extremum problem formulation in mixed form for linearly elastic structures is available as a special case as well.

Buckling Analysis of a Bi-Directional Strain-Gradient Euler–Bernoulli Nano-Beams

2020 ◽  
Vol 20 (11) ◽  
pp. 2050114
Author(s):  
Murat Çelik ◽  
Reha Artan
Keyword(s):  
Potential Energy ◽  
Gradient Elasticity ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Minimum Potential Energy ◽  
End Conditions ◽  
Graded Material ◽  
Minimum Potential ◽  
Gradient Elasticity Theory ◽  
Basic Equations

Investigated herein is the buckling of Euler–Bernoulli nano-beams made of bi-directional functionally graded material with the method of initial values in the frame of gradient elasticity. Since the transport matrix cannot be calculated analytically, the problem was examined with the help of an approximate transport matrix (matricant). This method can be easily applied with buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on gradient elasticity theory. Basic equations and boundary conditions are derived by using the principle of minimum potential energy. The diagrams and tables of the solutions for different end conditions and various values of the parameters are given and the results are discussed.

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