scholarly journals Static Mantle Density Distribution 3 Dimpling and Bucking of Spherical Crust

2020 ◽  
pp. 21-38
Author(s):  
Tian-Quan Yun
Keyword(s):  
Density Distribution ◽  
Turning Point ◽  
Unit Length ◽  
Internal Force ◽  
Governing Equations ◽  
Numerical Result ◽  
Analysis Question ◽  
Eigen Value ◽  
Non Linear ◽  
Distribution Equation

This paper is the third step of project “Static mantle distribution, Equation, Solution and Application”. It consists of < Static Mantle Distribution 1 Equation>, <Static Mantle Density Distribution 2 Improved Equation and Solution>, and this paper. Our result on shape of core is a “X type”, which differs from the traditional view that core is a sphere. Which one is correct? or, both are not correct? The aim of thispaper is to study dimpling and bucking of the spherical crust under mantle loading. Dimpling analysis depends on the outer solution of non-homogeneous non-linear D.E., while bucking analysis depends on non-linear Eigen value of the homogeneous D. E The results based on two models and governing equations show that crust dimpled at poles is proved theoretically and numerical result well consists with pole radius, while the non-linear bucking Eigen value boundary problem is solved by decomposition method. The results show that bucking can occur, and the un-continuity of internal force per unit length causes un-continuity of masses by mantle material emitting to crust at turning point of “X”. The growing of Tibet high-land might be viewed as an evidence of the mass ms(θ0)increasing due to mantle emission. Both poles radius and equatorial radius have been used to support our analysis. Question: how the nature makes cold at poles?

scholarly journals Non-linear buckling analysis of functionally graded shallow spherical shells

2009 ◽  
Vol 31 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Dao Huy Bich
Keyword(s):  
External Pressure ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Power Law Distribution ◽  
Governing Equations ◽  
Spherical Shells ◽  
Buckling Loads ◽  
Linear Buckling ◽  
Non Linear ◽  
Linear Buckling Analysis

In the present paper the non-linear buckling analysis of functionally graded spherical shells subjected to external pressure is investigated. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents of the material. In the formulation of governing equations geometric non-linearity in all strain-displacement relations of the shell is considered. Using Bubnov-Galerkin's method to solve the problem an approximated analytical expression of non-linear buckling loads of functionally graded spherical shells is obtained, that allows easily to investigate stability behaviors of the shell.

scholarly journals Non-linear analysis of laminated composite doubly curved shallow shells

2006 ◽  
Vol 28 (1) ◽  
pp. 43-55
Author(s):  
Dao Huy Bich
Keyword(s):  
Laminated Composite ◽  
Governing Equations ◽  
Shallow Shells ◽  
Approximate Analytical Solutions ◽  
Linear Buckling ◽  
Non Linear ◽  
Vibration Problems ◽  
Doubly Curved ◽  
Analytical Expressions ◽  
Load Deflection Curve

This paper deals with governing equations and approximate analytical solutions based on some wellknown assumptions to the non-linear buckling and vibration problems of laminated composite doubly curved shallow shells. Obtained results will be presented by analytical expressions of the lower critical load, the postbuckling load-deflection curve and the fundamental frequency of non-linear free vibration of the shell.

Transient Response of Inductrack Systems for Maglev Transport: Part II—Solution and Dynamic Analysis

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Ruiyang Wang ◽  
Bingen Yang
Keyword(s):  
Steady State ◽  
Dynamic Analysis ◽  
Transient Response ◽  
Numerical Procedure ◽  
Steady State Response ◽  
Governing Equations ◽  
Transient Model ◽  
State Response ◽  
Proposed Model ◽  
Non Linear

Abstract In Part I of this two-part paper, a new benchmark transient model of Inductrack systems is developed. In this Part II, the proposed model, which is governed by a set of non-linear integro-differential governing equations, is used to predict the dynamic response of Inductrack systems. In the development, a state-space representation of the non-linear governing equations is established and a numerical procedure with a specific moving circuit window for transient solutions is designed. The dynamic analysis of Inductrack systems with the proposed model has two major tasks. First, the proposed model is validated through comparison with the noted steady-state results in the literature. Second, the transient response of an Inductrack system is simulated and analyzed in several typical dynamic scenarios. The steady-state response results predicted by the new model agree with those obtained in the previous studies. On the other hand, the transient response simulation results reveal that an ideal steady-state response can hardly exist in those investigated dynamic scenarios. It is believed that the newly developed transient model provides a useful tool for dynamic analysis of Inductrack systems and for in-depth understanding of the complicated electro-magneto-mechanical interactions in this type of dynamic systems.

scholarly journals Mixed convection flow of a nanofluid past a non-linearly stretching wall

2018 ◽  
Vol 22 (1 Part B) ◽  
pp. 567-575
Author(s):  
Adeel Ahmad ◽  
Saleem Asghar ◽  
Mudassar Jalil ◽  
Ahmed Alsaedi
Keyword(s):  
Vertical Wall ◽  
Similarity Solutions ◽  
Numerical Results ◽  
Percentage Error ◽  
Convection Flow ◽  
Governing Equations ◽  
Keller Box Method ◽  
Group Theoretic Method ◽  
Non Linear ◽  
Mixed Convective Flow

This paper deals with the boundary-layer mixed convective flow of a viscous nanofluid past a vertical wall stretching with non-linear velocity. The governing equations are transformed into self similar ordinary differential equations using appropriate transformation. Using group theoretic method it is shown that the similarity solutions are possible only for the non-linear stretching velocity having specific form. Numerical solution of the coupled governing equations is obtained using Keller Box method. Correlation expression of reduced Nusselt and Sherwood numbers are obtained by performing linear regression on the data obtained from numerical results. The authenticity of these results is established by calculating the percentage error between the numerical results and correlation expression which is observed to be less than 5%. Effects of Brownian and thermophoretic diffusions and nanoparticles concentration flux on the Nusselt and Sherwood numbers are discussed.

scholarly journals Hydromagnetic flow of a Carreau fluid in a curved channel with non-linear thermal radiation

2019 ◽  
Vol 23 (6 Part A) ◽  
pp. 3379-3390 ◽  
Author(s):  
Zaheer Abbas ◽  
Muhammad Imran ◽  
Muhammad Naveed
Keyword(s):  
Thermal Radiation ◽  
Radiation Effects ◽  
Convective Boundary Condition ◽  
Weissenberg Number ◽  
Curved Channel ◽  
Convective Boundary ◽  
Flow Equations ◽  
Numerical Result ◽  
Non Linear ◽  
Rosseland Approximation

The study depicts the variations in the hydromagnetics flow of a Careau fluid in a semi permeable curved channel with convective boundary condition. Furthermore, Rosseland approximation is also considered to analyze the non-linear thermal radiation effects. Curvilinear co-ordinates system has been adopted for the mathematical modeling of the flow equations. The attained set of governing equation are then converted into non-linear dimensionless differential equations, by making use of similarity variables which are later treated by shooting method. In addition, the Newton?s Raphson method is also incepted to improve the accuracy of the obtained numerical result. The velocity field and temperature distributions are affected by various involved parameter which are presented in graphs and in table form. It is noticed that the velocity profiles are influenced by the change in the Weissenberg number.

scholarly journals Dimensionless characterization of the non-linear soil consolidation problem of Davis and Raymond. Extended models and universal curves

2019 ◽  
Vol 4 (1) ◽  
pp. 61-78 ◽  
Author(s):  
G. García-Ros ◽  
I. Alhama ◽  
F. Alhama
Keyword(s):  
Characteristic Time ◽  
Average Degree ◽  
Dimensionless Time ◽  
Soil Consolidation ◽  
Governing Equations ◽  
Dimensionless Groups ◽  
Non Linear ◽  
Consolidation Model ◽  
Loading Ratio

AbstractThe dimensionless groups that govern the Davis and Raymond non-linear consolidation model, and its extended versions resulting from eliminating several restrictive hypotheses, were deduced. By means of the governing equations nondimensionalization technique and introducing the characteristic time concept, both in terms of settlement and pressures, was obtained (for the most general model) that the average degree of settlement only depends on the dimensionless time while the average degree of pressure dissipation does it, additionally, on the loading ratio. These results allowed the construction of universal curves expressing the solutions of the unknowns of interest in a direct and simple way.

On Local and Global Species Conservation Errors for Nonlinear Ecological Models and Chemical Reacting Flows

2015 ◽  
Author(s):  
M. K. Mudunuru ◽  
M. Shabouei ◽  
K. B. Nakshatrala
Keyword(s):  
Numerical Solutions ◽  
Species Conservation ◽  
Chemical Species ◽  
Chemical Kinetic ◽  
Benchmark Problems ◽  
Governing Equations ◽  
Non Linear ◽  
Chemically Reacting Systems ◽  
Oscillatory Chemical ◽  
Reacting Systems

Advection-controlled and diffusion-controlled oscillatory chemical reactions appear in various areas of life sciences, hydrogeological systems, and contaminant transport. In this conference paper, we analyze whether the existing numerical formulations and commercial packages provide physically meaningful values for concentration of the chemical species for two popular oscillatory chemical kinetic schemes. The first one corresponds to the chlorine dioxide-iodine-malonic acid reaction while the second one is a simplified version of Belousov-Zhabotinsky reaction of a non-linear chemical oscillator. The governing equations for species balance are presented based on the theory of interacting continua. This results in a set of coupled non-linear partial differential equations. Obtaining analytical solutions is not practically viable. Moreover, it is well-known in literature that if the local dynamics becomes complex, the range of possible dynamic behavior in the presence of diffusion and advection becomes practically unlimited. We resort to numerical solutions, which are obtained using two popular stabilized formulations: Streamline Upwind/Petrov Galerkin and Galerkin/Least Squares. In order to make the computational analysis tractable, an estimate on the range of system-dependent parameters is obtained based on model reduction performed on the strong-form of the governing equations. Finally, we quantify the errors in satisfying the local and global species balance for various realistic benchmark problems. Through these representative numerical examples, we shall demonstrate the need and importance of developing locally conservative non-negative numerical formulations for chaotic and oscillatory chemically reacting systems.

Riemannian and Euclidean material structures in anelasticity

2020 ◽  
Vol 25 (6) ◽  
pp. 1267-1293 ◽  
Author(s):  
Fabio Sozio ◽  
Arash Yavari
Keyword(s):  
Geometric Structure ◽  
Deformation Gradient ◽  
Linear Momentum ◽  
The Other ◽  
Governing Equations ◽  
Multiplicative Decomposition ◽  
Material Forces ◽  
Geometric Structures ◽  
Non Linear

In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

Numerical Simulation of Water Impact of 2D Wedge with Different Density

2014 ◽  
Vol 623 ◽  
pp. 73-77
Author(s):  
Cheng Jun Li ◽  
Hui Long Ren ◽  
Chen Feng Li
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Linear Problem ◽  
Water Impact ◽  
Numerical Result ◽  
Non Linear

It remains unresolved to study the relation between the density of wedge and its impact. Fluent is applied to simulate its mechanism with 6DOF model. After the comparison of numerical result and the theoretical, it matches good and then the transient free surface scene is observed. Thus Fluent is able to solve this kind of non-linear problem. And the effect of density of wedge on the slamming character is also studied.

A Non-Linear Model for Stability Analysis of Reticulated Shallow Shells with Imperfections

1995 ◽  
Vol 10 (4) ◽  
pp. 215-230 ◽  
Author(s):  
G.H. Nie ◽  
Y.K. Cheung
Keyword(s):  
Virtual Work ◽  
Equivalent Model ◽  
Radius Of Curvature ◽  
Governing Equations ◽  
Shallow Shells ◽  
Coupled Equations ◽  
Linear Characteristic ◽  
Non Linear ◽  
Linear Stability Problem ◽  
Imperfection Factor

A non-linear stability problem of imperfect reticulated shallow shells with distorted rectangular meshes is investigated in this paper. The fundamental governing equations are deduced by adopting an equivalent model and the principle of the virtual work. For the reticulated shallow spherical shell under uniform vertical load, an axisymmetrical case is considered and the analytical solution of the coupled equations is given with the help of the asymptotical iteration method. Meanwhile, non-linear characteristic relations concerning load, deflection and imperfection factor are numerically analyzed. In particular, the corresponding solution degenerates to that of a reticulated circular plate when the radius of curvature of the structure R → ∞.

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