On finite-element solution of spatial thermoviscoelastoplastic problems

Lelyukh, Yu. I.; Shevchenko, Yu. N.

doi:10.1007/s10778-006-0113-0

Cited by 8 publications

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References 8 publications

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“…The system of governing equations (3.9) is solved by a numerical time-stepping method and by the method of successive approximations at each step, satisfying prescribed boundary conditions [5][6][7][8]. An algorithm to solve this system of equations may be the following.…”

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Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

2007

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Constitutive equations relating the components of the stress tensor in a Eulerian coordinate system and the linear components of the finite-strain tensor are derived. These stress and strain measures are energy-consistent. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator and that the first invariants of the stress and strain tensors are in a nonlinear relationship. In the case of combined elastoplastic deformation of elements of the body, this relationship, as well as the relationship between the second invariants of the stress and strain deviators, is determined from fundamental tests on a tubular specimen subjected to proportional loading at several values of stress mode angle (the third invariant of the stress deviator). Methods to individualize these relationships are proposed. The initial assumptions are experimentally validated. The constitutive equations derived underlie an algorithm for solving boundary-value problems Keywords: isotropic body, stress tensor, linear components of finite-strain tensor, Eulerian coordinates, elastoplastic deformation, path of small curvature, stress mode angle, constitutive equations Introduction. In deriving the constitutive equations describing the elastoplastic deformation of elements of an isotropic solid along paths of small curvature, we [9] accounted for the stress mode and made and experimentally validated certain assumptions. In these equations, the stress components are defined in a Eulerian coordinate system fixed at one point to the body and remaining unchanged during its deformation. The strain components defined in the same coordinate system are considered finite, i.e., the strain tensor has linear and nonlinear components. As shown in [10], the sum of products of these stress components and the variations of the finite-strain components is not the work done by forces during displacement variation-this work is equal to the sum of products of these stress components and the variations of not the total finite-strain components, but their linear parts only. We will examine the effect of the energy inconsistency of the stress and strain measures [2,9] on the reliability of the assumptions made to derive the constitutive equations [9]. To this end, we will validate the assumptions on the basis of fundamental tests for the energy-consistent stress and strain measures, taking the stress mode of the material into account.1. Constitutive Equations. To establish the relationship between the stress components and the linear finite-strain components in a Eulerian coordinate system, we will represent these tensors s ij and e ij as sums of deviatoric (s ij and e ij ) and spherical (a s 0 and e 0 ) components:

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Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

2007

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“…To solve the three-dimensional problem of thermoviscoelastoplasticity for a damaged material, we will use the widely known Newton-Raphson method [28]. According to this method, we need to formulate a problem for increments of the unknowns.…”

Section: Finite-element Algorithm For Solving Three-dimensional Problmentioning

confidence: 99%

“…For discretization purposes, we choose a hexahedral eight-node finite element [7,8]. We will solve the problem for displacements, following the technique detailed in [28]. After the total strains are determined at the current approximation, the irreversible strain increments (1.5) can be found.…”

Section: Finite-element Algorithm For Solving Three-dimensional Problmentioning

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“…In each approximation of each step, the elastic strain tensor e ij e is calculated as the difference between the total strain and the irreversible strain accumulated over the previous steps (1.5 [28]): where the coefficient p and the form of the function f ( ) w are determined from the best fit condition for the third stage of the creep curves using the technique proposed in [3,25]. Also, the above two methods can be combined to give a mixed method, namely:…”

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Analyzing methods to allow for the damage of material in thermoviscoelastoplastic deformation

Lelyukh

2007

Int Appl Mech

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Three methods to allow for damage of isotropic materials are discussed. The relations of the theory of deformation along paths of small curvature are used as equations of state. Rabotnov's scalar equation is used to study the damage of a material during thermoviscoelastoplastic deformation. The stress determined by a stress rupture criterion that accounts for the stress mode is taken as an equivalent stress. An algorithm based on the finite-element method is developed to solve three-dimensional problems of thermoviscoelastoplasticity with allowance for material damage. The numerical results obtained are compared with experimental data Keywords: thermoviscoelastoplasticity, damage, finite-element method, three-dimensional problem Introduction. The prediction of the total and residual service lifes of responsible members in modern structures operating for a long time under combined thermomechanical loading has recently come into current importance. This is because depending on thermal and mechanical conditions, plastic and creep strains may develop and stresses relax in such structures. Irreversible strains occurring under long-term loading may ultimately result in collapse of the whole structure.Phenomenological models allowing for the damage of a material under thermoviscoelastoplastic deformation and describing the kinetics of damage under quasistatic loading are addressed in [9,10,[13][14][15].These damage models were further developed in [1, 3, 4-6, 21, 23-27, 29-31]. The majority of these studies are based on various kinetic equations and damage parameters.According to Kachanov [9, 10] and Rabotnov [13][14][15], the key idea of phenomenological continuum models is to introduce so-called effective stresses dependent on some parameter describing the damage of the material. In the majority of the cited studies, effective stresses implicitly appear as a function of a damage parameter in the kinetic equation for the creep strain rate, which is reduced to modeling transient creep. Such models may be named the method of effective creep strains.A different approach was proposed in [21,24]. It directly allows for effective stresses by introducing effective moduli of elasticity depending on the damage of the material. This approach (we will call it the method of effective moduli) allows phenomenological description of the load-bearing capacity of the material as its damage builds up during deformation. Moreover, this method permits simple generalization to other types of damage such as high-cycle one.Generalizing the methods of effective creep strains and effective moduli, we can propose a mixed method describing both the variation in the mechanical characteristics of the material and transient creep. The present paper analyzes such methods against experimental data.1. Formulation of Three-Dimensional Thermoviscoplastic Problems for Damaged Materials. Consider, in curvilinear coordinates (q 1 , q 2 , q 3 ), a piecewise-inhomogeneous multilayer solid of volume V bounded by a surface S. At time zero, the solid is in natu...

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“…The technique is based on the geometrically nonlinear equations of the theory of thin shells derived assuming small tensile and shear strains and incorporating transverse-shear strains and torsions [7] and on equations of state describing the deformation of the body's element along paths of small curvature [14,15,21,22].…”

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Axisymmetric thermoviscoelastoplastic state of thin flexible shells with damages

Galishin

2008

Int Appl Mech

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The paper presents a technique to determine the axisymmetric geometrically nonlinear thermoviscoelastoplastic state of thin shells with damages. The technique is based on the geometrically nonlinear equations that incorporate transverse-shear strains. The equations of thermoelasticity that describe the deformation of the body's element along paths of small curvature are used as equations of state. The equivalent stress in the kinetic equations of damage and creep is determined from a failure criterion that accounts for the stress mode. As an example, the geometrically nonlinear thermoviscoelastoplastic deformation of a corrugated shell is analyzed and the time to its failure is determined Keywords: shells of revolution, thermoviscoplasticity, geometric nonlinearity, damage Introduction. Structures made of isotropic materials to serve under nonisothermal loading are widely used in engineering. While in service, they may accumulate plastic and creep strains, which may cause failure. The stress-strain state (SSS) of structural members was determined in [1-3, 5-8, 12, 16-21, etc.] taking into account irreversible strains. The damage of the material of shells of revolution was accounted for in [1-3, 6, 16, 17] of which [1, 2, 6] addressed geometrically linear thermoviscoplastic problems.We will outline a technique to determine the axisymmetric thermoviscoelastoplastic geometrically nonlinear SSS of thin flexible shells of revolution with damages. The technique is based on the geometrically nonlinear equations of the theory of thin shells derived assuming small tensile and shear strains and incorporating transverse-shear strains and torsions [7] and on equations of state describing the deformation of the body's element along paths of small curvature [14,15,21,22].1. Problem Formulation. Consider a shell of revolution with arbitrary meridian and meridionally varying thickness h. The position of an arbitrary point of the shell is described by curvilinear coordinates s, j, and V that are orthogonal before deformation (s ( ) s s s n 0 £ £ is the arc length of the coordinate meridian; j is the circumferential coordinate; and V ( ) V £ V £ V

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On finite-element solution of spatial thermoviscoelastoplastic problems

Cited by 8 publications

References 8 publications

Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

Analyzing methods to allow for the damage of material in thermoviscoelastoplastic deformation

Axisymmetric thermoviscoelastoplastic state of thin flexible shells with damages

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