scholarly journals Algorithms for Solving Boundary Value Problems of Deformable Solid Mechanics in View of Creep Strain

2018 ◽  
pp. 1-14 ◽  
Author(s):  
I. V. Stankevich ◽  
S. S. Volkov
Keyword(s):  
Boundary Value Problems ◽  
Initial Stress ◽  
Creep Strain ◽  
Solid Mechanics ◽  
Boundary Value ◽  
Point Of View ◽  
Operating Conditions ◽  
Main Trend ◽  
Deformable Solid ◽  
Initial Strains

Further development of power and, primarily, engine engineering is associated with significantly increasing specific indicators. For example, the main trend in development of gas turbine engines is to increase gas parameters before the turbine. At the same time, there is an intensive growth of thermal and mechanical tension, and first of all this applies to the parts and components of the flow range. The destruction of these structural elements may have grave consequences. Increasing reliability and durability of responsible components of engines under operating conditions of complex cyclic thermo-mechanical loading is one of the priority tasks of modern engine engineering.One of the factors to determine a design performance is high-temperature creep. When solving the problems of deformable solid mechanics (DSM) in terms of creep, various options of the theory of hereditary creep and three main technical theories of aging, flow and hardening are widely used. There are also theories known that use an apparatus of the structural models and mechanical analogues to describe the creep. Most theories satisfactorily describe the creep strain under constant or slowly changing loads. Analysis of stress-strain states under variable loads is better described by the theories of flow and hardening, and the theory of hardening has some advantages over the theories of flow, as it gives more exact approximation for experiment results. From the point of view of the computing cycle arrangement, the technical theories have well-known advantages over the hereditary ones.When using the finite element method (FEM) to solve the boundary value problems of DSM considering the creep strain, an explicit or implicit Euler scheme is very often used. Depending on the features of the problem under consideration, a solution algorithm is constructed either in accordance with the method of initial stress, or by the method of initial strains. The method of initial strains when solving the problems in terms of creep is used more often, because the application of an initial stress method for this class of problems is technically much more complicated. The paper examines the explicit and implicit Euler schemes in combination with FEM. Both schemes are formulated in accordance with the method of initial strains. A constitutive relation was chosen in the form of the theory of flows.

scholarly journals On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Hussein A. H. Salem ◽  
Mieczysław Cichoń
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Point Of View ◽  
Integral Boundary ◽  
Special Cases

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

A New Method for Nonlinear Two-Point Boundary Value Problems in Solid Mechanics

2001 ◽  
Vol 68 (5) ◽  
pp. 776-786 ◽  
Author(s):  
L. S. Ramachandra ◽  
D. Roy
Keyword(s):  
Boundary Value Problems ◽  
Vector Fields ◽  
Solid Mechanics ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Post Buckling ◽  
Independent Variable ◽  
Solution Parameters

A local and conditional linearization of vector fields, referred to as locally transversal linearization (LTL), is developed for accurately solving nonlinear and/or nonintegrable boundary value problems governed by ordinary differential equations. The locally linearized vector field is such that solution manifolds of the linearized equation transversally intersect those of the nonlinear BVP at a set of chosen points along the axis of the only independent variable. Within the framework of the LTL method, a BVP is treated as a constrained dynamical system, which in turn is posed as an initial value problem. (IVP) In the process, the LTL method replaces the discretized solution of a given system of nonlinear ODEs by that of a system of coupled nonlinear algebraic equations in terms of certain unknown solution parameters at these chosen points. A higher order version of the LTL method, with improved path sensitivity, is also considered wherein the dimension of the linearized equation needs to be increased. Finally, the procedure is used to determine post-buckling equilibrium paths of a geometrically nonlinear column with and without imperfections. Moreover, deflections of a tip-loaded nonlinear cantilever beam are also obtained. Comparisons with exact solutions, whenever available, and other approximate solutions demonstrate the remarkable accuracy of the proposed LTL method.

scholarly journals Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
J. Gwinner
Keyword(s):  
Variational Inequalities ◽  
Saddle Point ◽  
Boundary Value Problems ◽  
Solid Mechanics ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Set Convergence ◽  
Nonsmooth Boundary ◽  
Mixed Variational Inequalities

The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.

scholarly journals On joint efficiency of composite anisotropic plate rigidly fixed along outside edges

2020 ◽  
Vol 19 (4) ◽  
pp. 304-309
Author(s):  
A. G. Akopyan
Keyword(s):  
Classical Theory ◽  
Solid Mechanics ◽  
Anisotropic Plate ◽  
Composite Plates ◽  
Local Stress ◽  
Operating Conditions ◽  
Structural Elements ◽  
Deformable Solid ◽  
Engineering Structures ◽  
Anisotropic Plates

Introduction. Modern processes of welding, surfacing, soldering and bonding provide producing structural elements of monolithic interconnected dissimilar anisotropic materials. The combination of different materials with qualities corresponding to certain operating conditions offer comprehensive facilities to improve the technical and economic characteristics of machines, equipment and structures. It can contribute to a significant increase in their reliability, durability, and to reduction of the production and operation costs. Materials and Methods. The work objective is to study the boundary state of stress of anisotropic composite plates in the framework of the classical theory of plate bending. The outer edges of the plate are considered free. Using the classical theory of bending of an anisotropic plate in the space of physical and geometric parameters, hypersurface equations are obtained that define low-stressed zones for the contact surface edge of a cylindrical orthotropic composite plate.Research Results. Finding the criteria for engineering structures to determine the limiting strength characteristics of structural elements is one of the urgent tasks of the deformable solid mechanics. Strength problems in structures are often reduced to elucidating the nature of the local stress state at the tops of the joints of the constituent parts. This paper is devoted to solving this problem for composite anisotropic plates in the area of their bending.Discussion and Conclusions. The solution proposed in this paper may be useful for increasing the strength of composite products.

scholarly journals Solution of the problems of quasi-statics for an elastic body with double porosity

2021 ◽  
Vol 1 (3) ◽  
pp. 1-5
Author(s):  
Ivane Tsagareli
Keyword(s):  
Boundary Value Problems ◽  
Elastic Body ◽  
Boundary Value ◽  
Original System ◽  
Double Porosity ◽  
Point Of View ◽  
Static Case ◽  
Elementary Functions ◽  
The Laplace Transform ◽  
Dimensional Boundary

The construction of solutions in explicit form is especially important from the point of view of its application, since it makes it possible to effectively carry out a quantitative analysis of the problem under study. This paper investigates the processes of deformation of solids in the quasi-static case. Two-dimensional boundary value problems of Dirichlet and Neumann for an elastic body with double porosity are considered. In Using the Laplace transform, these problems are reduced to auxiliary boundary value problems. Special representations of solutions to auxiliary boundary value problems are constructed using elementary functions that allow reducing the original system of equations to equations of a simple structure and facilitate the solution of the original problems. Auxiliary boundary value problems are solved for a specific elastic body - a porous disk. Solutions to these problems are obtained in the form of series. Conditions are provided that ensure the absolute and uniform convergence of these series and the use of the inverse Laplace theorem. It is proved that the inverse transforms provide a solution to the initial problems.

Variational formulation of boundary-value problems

2020 ◽  
pp. 215-222
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Value Problems ◽  
Variational Formulation ◽  
Solid Mechanics ◽  
Boundary Value ◽  
Small Deformation ◽  
Local Form ◽  
Virtual Power ◽  
Cauchy Relation ◽  
Principle Of Virtual Power ◽  
Balance Of Forces

This chapter discusses a variational formulation of boundary value problems in small deformation solid mechanics. It begins by introducing the important principle of virtual power, and shows that it encapsulates Cauchy’s traction law, and the local form of the basic balance of forces (equation of equilibrium), and the local from of the balance of moments (symmetry of the stress). Since the principle of virtual power encapsulates both the equation of equilibrium and the Cauchy relation for tractions, it can be used to formulate and solve boundary-value problems in solid mechanics in a variational or weak sense. Specifically, it is shown how the displacement problem of linear elastostatics may be formulated variationally using the principle of virtual power.

scholarly journals Variational methods for boundary value problems

2000 ◽  
Vol 4 (2) ◽  
pp. 193-204
Author(s):  
B. Tsang ◽  
S. W. Taylor ◽  
G. C. Wake
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Point Of View ◽  
One Dimension ◽  
Computational Point ◽  
Higher Dimensional ◽  
Special Cases ◽  
Variational Form ◽  
General Boundary Value Problem

The variational formulation of boundary value problems is valuable in providing remarkably easy computational algorithms as well as an alternative framework with which to prove existence results. Boundary conditions impose constraints which can be annoying from a computational point of view. The question is then posed: what is the most general boundary value problem which can be posed in variational form with the boundary conditions appearing naturally? Special cases of two-point problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjointness for the linear case. Further cases under which a Lagrangian may or may not exist are explained.

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