scholarly journals The Hamiltonian System Method for the Stress Analysis in Axisymmetric Problems of Viscoelastic Solids

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
W. X. Zhang ◽  
Y. Bai ◽  
F. Yuan
Keyword(s):  
Hamiltonian System ◽  
Boundary Conditions ◽  
Integral Transformation ◽  
Separation Of Variables ◽  
Expansion Method ◽  
Stress Concentrations ◽  
Viscoelastic Solids ◽  
Quasistatic Problem ◽  
General Solutions ◽  
Various Boundary Conditions

With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.

The Symplectic System for Thermo-Viscoelastic Cylinders

2011 ◽  
Vol 250-253 ◽  
pp. 3662-3665
Author(s):  
Wei Xiang Zhang ◽  
Qi Xia Liu
Keyword(s):  
Boundary Conditions ◽  
Separation Of Variables ◽  
Solution Space ◽  
Governing Equations ◽  
Laplace Domain ◽  
General Solutions ◽  
Various Boundary Conditions ◽  
Temperature Boundary ◽  
The Time Domain ◽  
Symplectic System

An exact symplectic approach is presented for the isotropic viscoelastic solids subjected to external force and temperature boundary conditions. With the use of the method of separation of variables, all the general solutions of the governing equations are derived in the Laplace domain. These general solutions are expressed in concise analytical forms, and are easily to be transformed into the time domain. Accordingly, various boundary conditions can be conveniently described by the combination of the general solutions due to the completeness of the solution space. In the numerical example, the whole character of total creep of the viscoelastic solid is clearly exhibited.

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

scholarly journals Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension

1952 ◽  
Vol 19 (4) ◽  
pp. 526-528
Author(s):  
M. L. Williams
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Poisson's Ratio ◽  
Vertex Angle ◽  
Stress Singularities ◽  
Stress Concentrations ◽  
Various Boundary Conditions ◽  
Free Case

Abstract As an analog to the bending case published in an earlier paper, the stress singularities in plates subjected to extension in their plane are discussed. Three sets of boundary conditions on the radial edges are investigated: free-free, clamped-clamped, and clamped-free. Providing the vertex angle is less than 180 degrees, it is found that unbounded stresses occur at the vertex only in the case of the mixed boundary condition with the strength of the singularity being somewhat stronger than for the similar bending case. For vertex angles between 180 and 360 degrees, all the cases considered may have stress singularities. In amplification of some work of Southwell, it is shown that there are certain analogies between the characteristic equations governing the stresses in extension and bending, respectively, if ν, Poisson’s ratio, is replaced by −ν. Finally, the free-free extensional plate behaves locally at the origin exactly the same as a clamped-clamped plate in bending, independent of Poisson’s ratio. In conclusion, it is noted that the free-free case analysis may be applied to stress concentrations in V-shaped notches.

Influence of Boundary Conditions on Decay Rates in a Prestrained Plate1

2002 ◽  
Vol 69 (4) ◽  
pp. 515-520 ◽  
Author(s):  
B. Karp ◽  
D. Durban
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Boundary Data ◽  
Separation Of Variables ◽  
Finite Elasticity ◽  
Decay Rates ◽  
Constitutive Behavior ◽  
Hyperelastic Materials ◽  
Various Boundary Conditions ◽  
Decay Exponent

Decay of end perturbations imposed on a prestrained semi-infinite rectangular plate is investigated in the context of plane-strain incremental finite elasticity. A separation of variables eigenfunction formulation is used for the perturbed field within the plate. Numerical results for the leading decay exponent are given for three hyperelastic materials with various boundary conditions at the long faces of the plate. The study exposes a considerable sensitivity of axial decay rates to boundary data, to initial strain and to constitutive behavior. It is suggested that the results are relevant to the applicability of Saint-Venant’s principle even though the eigenfunctions are not always self-equilibrating.

Finite element modelling of sandwich panels with graded core under various boundary conditions

2012 ◽  
Vol 116 (1186) ◽  
pp. 1289-1314 ◽  
Author(s):  
M. Kashtalyan ◽  
B. Woodward
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Sandwich Panels ◽  
Stress Concentrations ◽  
Displacement Fields ◽  
Various Boundary Conditions ◽  
The Core ◽  
The Face ◽  
3D Elasticity ◽  
3D Finite Element Method

Abstract Sandwich panels are widely used in the aerospace industry instead of solid plates due to their high flexural stiffness-to-weight and flexural strength-to-weight ratios. However due to the mismatch of properties between the face sheets and the core, stress concentrations can occur at the face sheet/core interfaces, often leading to delamination. One possible solution to this problem is the introduction of a graded core — a core in which the properties vary gradually from the face sheets to the core centre, eliminating any abrupt changes in properties. In this paper a 3D finite element method, fully validated through comparison with results from the literature and a 3D elasticity solution, is applied to modelling of sandwich panels with graded core. The approach makes use of graded elements to study the effect of varying the boundary conditions on the elastic deformation of the panel subject to uniformly distributed loading. Comparative analysis of stress and displacement fields in sandwich panels with homogeneous and graded cores is carried out under various combinations of simply supported, clamped and free edges.

scholarly journals Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems

Pharmaceutics ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 140 ◽  
Author(s):  
Constantin Mircioiu ◽  
Victor Voicu ◽  
Valentina Anuta ◽  
Andra Tudose ◽  
Christian Celia ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Moving Boundary ◽  
Initial Conditions ◽  
Integral Transformation ◽  
Release Kinetics ◽  
Separation Of Variables ◽  
Transformation Method ◽  
Mathematical Methods ◽  
Important Place ◽  
First Choice

Embedding of active substances in supramolecular systems has as the main goal to ensure the controlled release of the active ingredients. Whatever the final architecture or entrapment mechanism, modeling of release is challenging due to the moving boundary conditions and complex initial conditions. Despite huge diversity of formulations, diffusion phenomena are involved in practically all release processes. The approach in this paper starts, therefore, from mathematical methods for solving the diffusion equation in initial and boundary conditions, which are further connected with phenomenological conditions, simplified and idealized in order to lead to problems which can be analytically solved. Consequently, the release models are classified starting from the geometry of diffusion domain, initial conditions, and conditions on frontiers. Taking into account that practically all solutions of the models use the separation of variables method and integral transformation method, two specific applications of these methods are included. This paper suggests that “good modeling practice” of release kinetics consists essentially of identifying the most appropriate mathematical conditions corresponding to implied physicochemical phenomena. However, in most of the cases, models can be written but analytical solutions for these models cannot be obtained. Consequently, empiric models remain the first choice, and they receive an important place in the review.

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

Electro-osmotic chemical behavior of clayey soil under various boundary conditions

2021 ◽  
Vol 28 (5) ◽  
pp. 1493-1504
Author(s):  
Zhi-jia Xue ◽  
Qi Xiong
Keyword(s):  
Boundary Conditions ◽  
Clayey Soil ◽  
Chemical Behavior ◽  
Various Boundary Conditions

Analytic solutions of the scattering by two multilayered dielectric spheres

1992 ◽  
Vol 70 (9) ◽  
pp. 696-705 ◽  
Author(s):  
A-K. Hamid ◽  
I. R. Ciric ◽  
M. Hamid
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Plane Electromagnetic Wave ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Addition Theorem ◽  
Modal Expansion ◽  
Modal Expansion Method ◽  
Dielectric Spheres ◽  
Scattered Fields

The problem of plane electromagnetic wave scattering by two concentrically layered dielectric spheres is investigated analytically using the modal expansion method. Two different solutions to this problem are obtained. In the first solution the boundary conditions are satisfied simultaneously at all spherical interfaces, while in the second solution an iterative approach is used and the boundary conditions are satisfied successively for each iteration. To impose the boundary conditions at the outer surface of the spheres, the translation addition theorem of the spherical vector wave functions is employed to express the scattered fields by one sphere in the coordiante system of the other sphere. Numerical results for the bistatic and back-scattering cross sections are presented graphically for various sphere sizes, layer thicknesses and permittivities, and angles of incidence.

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