TY - JOUR

T1 - Modified formulation of the iterative algorithm for solving linear viscoelasticity problems based on the separation of time and space variables

AU - Pavlov, M. S.

AU - Svetashkov, A. A.

AU - Kupriyanov, N. A.

N1 - Publisher Copyright:
© 2019 Tomsk State University. All rights reserved.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - Designing of the structures made of polymeric and composite viscoelastic materials requires development of the efficient and cost-effective methods for calculating stress-strain state. This paper proposes an iterative algorithm for solving such problems. The advantages of the algorithm over existing methods are the following: firstly, there is a possibility to parallelize the calculations of spatial and time components of the stress-strain state; secondly, this algorithm eliminates the need to integrate the history of stress and displacement variation in time. Moreover, the formulated iterative algorithm allows one to obtain the parameters of the stress-strain state of a viscoelastic solid without using integral operator inverse to the relaxation operator. The proposed method involves the following concept. The integral operators of the shear and volume relaxation are replaced by some values of the elastic shear and volume moduli. The identity of the obtained elastic problem with initially stated viscoelastic problem is ensured by supplementing right-hand sides of the equilibrium equations and boundary conditions with the corresponding residuals. In addition, each residual involves the result of viscoelastic operator effect on the required parameters, and, therefore, it cannot be found directly. The numerical implementation assumes the iteration process to be built, in which the residuals on the current step are calculated using the solutions obtained on the previous one. The paper describes the formulated iterative algorithm, as well as its application in conjunction with commercial or free computer software employed for a finite element analysis. The paper also includes an example of the model problem solution.

AB - Designing of the structures made of polymeric and composite viscoelastic materials requires development of the efficient and cost-effective methods for calculating stress-strain state. This paper proposes an iterative algorithm for solving such problems. The advantages of the algorithm over existing methods are the following: firstly, there is a possibility to parallelize the calculations of spatial and time components of the stress-strain state; secondly, this algorithm eliminates the need to integrate the history of stress and displacement variation in time. Moreover, the formulated iterative algorithm allows one to obtain the parameters of the stress-strain state of a viscoelastic solid without using integral operator inverse to the relaxation operator. The proposed method involves the following concept. The integral operators of the shear and volume relaxation are replaced by some values of the elastic shear and volume moduli. The identity of the obtained elastic problem with initially stated viscoelastic problem is ensured by supplementing right-hand sides of the equilibrium equations and boundary conditions with the corresponding residuals. In addition, each residual involves the result of viscoelastic operator effect on the required parameters, and, therefore, it cannot be found directly. The numerical implementation assumes the iteration process to be built, in which the residuals on the current step are calculated using the solutions obtained on the previous one. The paper describes the formulated iterative algorithm, as well as its application in conjunction with commercial or free computer software employed for a finite element analysis. The paper also includes an example of the model problem solution.

KW - auxiliary constitutive equations

KW - convergence

KW - integral operators

KW - iterative algorithm

KW - linear viscoelasticity

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U2 - 10.17223/19988621/61/8

DO - 10.17223/19988621/61/8

M3 - Article

AN - SCOPUS:85075339609

VL - 2019

SP - 82

EP - 94

JO - Vestnik Tomskogo Gosudarstvennogo Universiteta, Matematika i Mekhanika

JF - Vestnik Tomskogo Gosudarstvennogo Universiteta, Matematika i Mekhanika

SN - 1998-8621

IS - 61

ER -