On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

Babeshko, M. E.; Shevchenko, Yu. N.

doi:10.1007/s10778-008-0082-6

Cited by 5 publications

(2 citation statements)

References 10 publications

(28 reference statements)

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“…To solve the problem, we will use the Kirchhoff-Love hypotheses and the geometrically nonlinear theory of shells, i.e., the equilibrium and kinematic equations in traditional form [4][5][6]. Step number N s * × -10 3 , N/m q V * , ÌPà…”

Section: Numerical Resultsmentioning

confidence: 99%

Method of successive approximations for solving boundary-value problems of plasticity with allowance for the stress mode

Babeshko

Shevchenko

2010

Int Appl Mech

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A method for solving boundary-value problems of plasticity with allowance for the stress mode is developed. To describe the elastoplastic deformation of an isotropic material, use is made of constitutive equations that include two nonlinear functions dependent on the stress mode and determined experimentally. The elastoplastic state of a thin cylindrical shell under axisymmetric loading is calculated as a numerical example. The numerical results demonstrate good convergence of the method. The effect of the stress mode on the strain distribution in a cylindrical shell is assessed Keywords: boundary-value problems of plasticity, stress mode, third stress invariant, thin cylindrical shell Introduction. Constitutive equations describing the elastoplastic deformation of isotropic materials along paths of small curvature were proposed and experimentally validated in [7,[10][11][12]. These equations relate the components of the stress tensors and linear strains and assume that the strains have elastic and plastic components and the stress deviators and plastic strain differentials are coaxial. The equations include two nonlinear functions determined experimentally, one relating the mean stress and mean strain to the stress mode parameter, while the other relating the shear-stress and shear-strain intensities to the same parameter. This parameter is taken to be the stress mode angle [2]. If we replace the former nonlinear function by a linear relation between the first invariants of the stress and strain tensors and assume that the latter function is independent of the stress mode and determined from simple-tension tests, then the constitutive equations go over into those describing deformation along paths of small curvature [4], which in the case of active loading coincide with the equations of incremental plasticity [2, 8, etc.] associated with the von Mises yield criterion.To linearize the constitutive equations [10-12], the method of additional stresses was used in [10], where the general process of successive approximations for solving boundary-value static problems was outlined. Simplified constitutive equations [10-12] were considered in [1], where the former nonlinear function was replaced by a linear relation between the mean stress and mean strain and the possibility of such a replacement was justified. The simplified constitutive equations were used in [9] to solve a spatial problem of plasticity for a body of revolution based on the method of successive approximations. Unlike [9] and further to [10], the present paper details the algorithm of successive approximations based on the simplified constitutive equations that include the two nonlinear functions mentioned above and allow for unloading. The algorithm will be tested by solving a problem for a thin shell under axisymmetric loading. The method will be analyzed for practical convergence. Problem Formulation and Governing Equations.Consider an isotropic body (its mechanical properties are similar in all directions, the stress mode parameter being constan...

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Section: Numerical Resultsmentioning

confidence: 99%

Method of successive approximations for solving boundary-value problems of plasticity with allowance for the stress mode

Babeshko

Shevchenko

2010

Int Appl Mech

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“…The theory of elastoplastic [1,5,10,[13][14][15]21] and other compound media allows for plastic strains. Plastic zones [16,17] cause stress redistribution.…”

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Development of instability in a rotating elastoplastic annular disk

Lila

Martynyuk

2012

Int Appl Mech

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The paper proposes a method, based on perfect-plasticity and perturbation theories, for instability analysis of an annular flat disk tightly set on a shaft with no interference fit. The perturbed elastoplastic state of the rotating disk is analyzed by determining the stress-strain state of a fixed elastic annular plate under in-plane loading. A characteristic equation of the first order for the critical radius of the plastic zone in the disk subject to internal pressure is derived. The critical rotation rate is calculated for different parameters of the disk Keywords: annular flat disk, axisymmetric elastoplastic problem, instability, rotating shaft, shape perturbation method, critical rotation rateIntroduction. The theory of elastoplastic [1,5,10,[13][14][15]21] and other compound media allows for plastic strains. Plastic zones [16,17] cause stress redistribution. The stress is maximum at the elastic-plastic boundary. In this connection, it is of special interest to locate the elastic-plastic boundary in elastoplastic problems.Plastic-equilibrium problems arise even when dealing with simplest stress-strain states in which the stresses and strains depend on one coordinate only. Among them are many problems of practical importance such as the equilibrium of cylindrical pipes and spherical vessels or the uniform rotation of cylinders and disks under high loads or at high angular rates. Since the yield point can be exceeded in some zones of very rapidly rotating cylinders or disks, it has long been recognized that it is necessity to use highly ductile materials for heavy shafts of steam turbines or massive cylindrical rotors of big turbogenerators, which are subjected mainly to centrifugal stresses [11].At high rotation rates, a plastic zone appears at the center of a solid disk or along the internal edge of a disk with a hole. As the rotation rate is increased, the plastic zone extends over the entire disk. This increase is approximately 12% for the solid disk [11]. However, the disk can no longer be used as before, even prior to the loss of load-carrying capacity [7,19,20] occurring when the plastic zone reaches the outer edge of the disk. This is because of the instability of the disk in elastoplastic state-it takes a flat equilibrium shape different from the initial circular shape. In [6-8, etc.], the corresponding critical rotation rate w * for disks made of an incompressible material was determined and the loss of load-carrying capacity of rotating solid disks was studied.Here we outline a method, based on the perturbation method [2][3][4]8], to determine the critical rotation rate for an elastoplastic homogeneous isotropic flat annular disk tightly set on a shaft. The radius of the shaft is equal to the radius of the hole in the disk. The inside surface of the disk is subject to the pressure exerted by the shaft during rotation.1. Problem Formulation. Small deviations from the circular shape of a disk and misaligned fit on the shaft are known to have an insignificant effect (to 3%) on the loss of its load-car...

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Using mesh-based methods to solve nonlinear problems of statics for thin shells

2009

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The paper outlines a numerical procedure for solving physically and geometrically nonlinear problems of statics for thin shells based on three mesh-based methods: finite-difference, variational difference, and finite-element methods. The methodological, algorithmic, and analytical aspects of implementing the Kirchhoff-Love hypotheses are analyzed. The algorithmic approach employs Lagrangian multipliers. The advantages and disadvantages of these methods are evaluated Keywords: finite-difference method, variational difference method, finite-element method, Kirchhoff-Love hypotheses, Lagrangian multipliers, nonlinear elastic composite, elastoplastic stateIntroduction. Strength analysis of structural members such as shells of complex geometry is mainly conducted on a computer using numerical mesh-based methods, including the finite-difference method (FDM), the variational difference method (VDM), and the finite-element method (FEM). To derive the basic equations of the theory of thin shells, it is sufficient to use the classical model based on the Kirchhoff-Love hypotheses. The Kirchhoff-Love kinematics is implemented differently in these three methods.The system of governing equations in the FDM is known to be derived from the equilibrium equations for an element of the shell, while the Kirchhoff-Love constraints are incorporated analytically, by substituting the corresponding relations into the formulas describing the distribution of displacements throughout the thickness of the shell. This way causes no complications. Contrastingly, the system of governing equations in the VDM and FEM is derived from the extremum condition for some functional. An attempt to implement Kirchhoff-Love constraints analytically gives rise to higher-than-first-order derivatives of displacements in the functionals. Note that the complications in the VDM are only restricted to more awkward equations and the necessity of deriving them because the VDM imposes no special requirements upon the smoothness of the integrands and functions. In the FEM, however, certain requirements are placed upon the smoothness of the coordinate functions, and the construction of an element is complicated if the integrands include higher-order derivatives.These circumstances led to a search for alternative ways of incorporating the Kirchhoff-Love constraints such as those that would not produce derivatives of higher than the first order in the integrands. Noteworthy are two groups of approaches that employ Lagrangian multipliers and penal functions. Applying them requires certain correctness [17,52] associated with the variation order and additional conditions. However, the use of Lagrangian multipliers to implement the Kirchhoff-Love hypotheses increases the number of varied functions and requires much computing. This is obviously why the Kirchhoff-Love constraints were first (1967) used [46] in the FEM for a plate by specifying them at discrete points of an element (discrete Kirchhoff element). Later, this method was extended to thin shells [45,70] and is wi...

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On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

Cited by 5 publications

References 10 publications

Method of successive approximations for solving boundary-value problems of plasticity with allowance for the stress mode

Method of successive approximations for solving boundary-value problems of plasticity with allowance for the stress mode

Development of instability in a rotating elastoplastic annular disk

Using mesh-based methods to solve nonlinear problems of statics for thin shells

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