Deformation due to surface temperature variation on a spherically layered, transversely isotropic and self-gravitating Earth

2021 ◽  
Author(s):  
Jiangcun Zhou ◽  
Ernian Pan ◽  
Michael Bevis
Keyword(s):  
Boundary Value Problem ◽  
Analytical Method ◽  
Thermal Load ◽  
Boundary Value ◽  
Transversely Isotropic ◽  
Earth Model ◽  
Eighth Order ◽  
Thermally Induced ◽  
Thermal Anisotropy ◽  
Love Numbers

Summary We present a theory of modern, thermally-induced deformation in a realistic Earth. The heat conduction equation is coupled with standard elastic deformation theory to construct a boundary-value problem comprised of eighth-order differential equations. The accurate and stable dual variable and position propagating matrix technique is introduced to solve the boundary-value problem. The thermal load Love numbers are defined to describe the displacements and potential changes driven by thermally-induced deformation. The proposed analytical method is validated by comparing the present results with exact solutions for a homogeneous sphere, which are also derived in this paper. The analytical method is then applied to a realistic Earth model to evaluate the effects of layering and self-gravitation of the Earth on displacement and changes of potential. Furthermore, the frequency-dependence in the thermal load is illustrated by invoking different thermal periodicities in the computation. Thermal anisotropy is also considered by comparing the results obtained using isotropic and transversely isotropic Earth models. Results show that, when simulating thermally-induced deformation, invoking a homogeneous spherical Earth leads to results that substantially differ from those obtained using a more realistic Earth model.

scholarly journals Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

2020 ◽  
Vol 22 (1) ◽  
pp. 48-70
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Boundary Value ◽  
Weight Functions ◽  
Order Differential Operator ◽  
Eighth Order ◽  
Entire Family ◽  
Indicator Diagram

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of "conjugation" must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of "conjugation" are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

Design of Streamline Dies for Drawing Driven by Fracture

2019 ◽  
Vol 295 ◽  
pp. 85-89
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Prashant P. Date
Keyword(s):  
Boundary Value Problem ◽  
Ductile Fracture ◽  
Analytical Method ◽  
Fracture Criterion ◽  
Boundary Value ◽  
Simple Relation ◽  
Geometric Parameters ◽  
Constitutive Parameter ◽  
Bernoulli’S Theorem

This paper presents an efficient analytical method for design of streamline dies driven by fracture. The method is based on Bernoulli’s theorem relating pressure and velocity along any streamline extended to ideal flows in plasticity. The Cockroft-Latham criterion is adopted to predict the initiation of ductile fracture. In order to apply the method developed, it is not necessary to know the solution to the boundary value problem of plasticity. The final result is a simple relation between geometric parameters of the process and the constitutive parameter involved in the fracture criterion. Since the latter is supposed to be known for a given material, the relation determines a safe domain for drawing without fracture.

scholarly journals Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray–Schauder Alternative Fixed Point Theorem

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 129 ◽  
Author(s):  
Thenmozhi Shanmugam ◽  
Marudai Muthiah ◽  
Stojan Radenović
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Eighth Order ◽  
Classical Version ◽  
Nonlinear Alternative

In this work, we investigate the existence of solutions for the particular type of the eighth-order boundary value problem. We prove our results using classical version of Leray–Schauder nonlinear alternative fixed point theorem. Also we produce a few examples to illustrate our results.

scholarly journals Numerical Modeling of the Process of Thermoplastic Deformation of Transversally Isotropic Parallelepipeds

2020 ◽  
Vol 9 (4) ◽  
pp. 1484-1491
Keyword(s):  
Finite Difference ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Displacement Vector ◽  
Boundary Value ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Discrete Analogue ◽  
First Approximation ◽  
Transversally Isotropic

The paper proposes a modified version of the iterative method for numerically solving a three-dimensional uncoupled boundary-value problem that describes the process of thermoplastic deformations of a transversely isotropic parallelepiped. A discrete analogue of the boundary value problem is compiled on the basis of the finite-difference method. A recurrent finite-difference relation is written which allows one to find the desired components of the displacement vector in combination with the iterative method. It is assumed that, at a first approximation, the values of the sought displacements in the internal nodes are trivial. The essence of the method is demonstrated by solving the thermoplastic boundary-value problem for a transversely isotropic parallelepiped. The proposed method can be applied to solve related problems of dynamic thermoplasticity.

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