Numerical methods in the theory of viscoelasticity

Pobedrya, B. E.

doi:10.1007/bf00856383

Cited by 7 publications

(19 citation statements)

References 1 publication

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“…Let's note that the finite-difference schemes of boundary value problems are symmetric. The convergence of iterative methods is provided within the framework of the generalized theorem on the convergence of iterative methods [24,26].…”

Section: Resultsmentioning

confidence: 99%

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A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

2023

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The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation. Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written. The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

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Section: Resultsmentioning

confidence: 99%

“…Similarly, to (26), the plane problem of elasticity theory can be formulated on the basis of the equilibrium equations ( 15), ( 16) and (19), i.e. :…”

Section: Engineeringmentioning

confidence: 99%

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

2023

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“…Currently, the numerical finite element method [8][9][10] is widely used to solve the problems of mechanics of a solid deformable body. This method allows approaching solutions of technical problems in the conditions close to practical and connected with the features of production.…”

Section: Methodsmentioning

confidence: 99%

The simulation of cold volumetric stamping by the method of transverse extrusion

2018

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The article is devoted to the theoretical study and development of the production process of manufacturing rod products with larger heads by transverse extrusion. For carrying out researches the elastic-plastic finite-element model based on the variation principle was chosen. This model, due to the development of a complex of boundary and initial conditions, has been adapted to the scheme of volume stamping of the fasteners and implemented in the form of a software package in the system DEFORM 3D. The paper presents the results of computer simulation of the technology of manufacturing the mortgage bolt

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“…Here not only constant coefficients can vary [37], but also whole functions and functionals such as the hardening function, the creep and relaxation functionals, etc. Instability can be interpreted as a change in the type of the boundary-value problem [40], which leads to qualitatively new properties of the solution. This is a source of many classical paradoxes in mechanics and engineering applications (limiting transition, destabilization, false resonances, etc.)…”

Section: Stability Of Deformation Processes Against Perturbations Of mentioning

confidence: 99%

Deformation of a Bingham viscoplastic fluid in a plane confuser

et al. 2006

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A review is given to and comprehensive numerical-analytic study is carried out of the problem of steady Bingham viscoplastic flow in a plane confuser. The solution is constructed in the first approximation with the yield stress as a small parameter and the solution of the Jeffery-Hamel problem (steady radial motion of an incompressible viscous material in a plane confuser) as the zero-order approximation. The numerical analysis is based on the modified accelerated-convergence method proposed earlier by the authors. The bifurcations of the deformation pattern occurring when the parameters reach some critical values are discussed and commented on. The asymptotic boundaries of the rigid zones that appear at infinity upon perturbation of the yield stress are determined Introduction. Recently, the journal Prikladnaya Mekhanika (International Applied Mechanics) has published review papers on solid mechanics and related subject areas [60, 63, 66, etc.], which is of practical interest for scientists who solve important problems based on recent achievements in mathematics and mechanics. This paper outlines and analyzes the theoretical results from solution of one specific problem.This problem describes the stationary deformation of a Bingham viscoplastic medium in a plane confuser. To solve it, we will use numerical and analytical methods. The solution will be found in the first approximation with the yield stress as a small parameter and the solution of the Jeffery-Hamel problem (steady radial motion of a viscous incompressible medium in a flat-walled converging channel) as the zero-order approximation. We will determine and discuss the main characteristics of the process-asymptotic (first-order) quasi-rigid and deformed zones and analyze their behavior over a wide range of parameters. New mechanical effects will be revealed (bifurcations in the deformation pattern, shape and position of the asymptotic quasi-rigid zones, and the behavior of the velocity of the medium at their boundaries).Moreover, we will also perform a complete numerical and analytic study of the Jeffery-Hamel problem over the whole range of dimensionless parameters-the opening angle of the confuser and the Reynolds number. The numerical analysis is based on the modified accelerated-convergence method proposed by the authors.

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Numerical methods in the theory of viscoelasticity

Cited by 7 publications

References 1 publication

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

The simulation of cold volumetric stamping by the method of transverse extrusion

Deformation of a Bingham viscoplastic fluid in a plane confuser

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