Thermoelastoplastic deformation of noncircular cylindrical shells

Merzlyakov, V. A.

doi:10.1007/s10778-008-0101-7

Int Appl Mech

2008

DOI: 10.1007/s10778-008-0101-7

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Thermoelastoplastic deformation of noncircular cylindrical shells

V. A. Merzlyakov

Abstract: A method to determine the nonstationary temperature fields and the thermoelastoplastic stress-strain state of noncircular cylindrical shells is developed. It is assumed that the physical and mechanical properties are dependent on temperature. The heat-conduction problem is solved using an explicit difference scheme. The temperature variation throughout the thickness is described by a power polynomial. For the other two coordinates, finite differences are used. The thermoplastic problem is solved using the geom… Show more

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Cited by 3 publications

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“…The influence of orthotropy on the stress-strain state in nonthin elliptic cylindrical shells is examined in [11]. The thermoelastoplastic deformation of noncircular cylindrical shells is studied in [12]. The free vibrations of three-layer elliptic cylindrical shells are addressed in [1].…”

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Forced vibrations of three-layer elliptic cylindrical shells under distributed loads

Meish

Mikhlyak

2010

Int Appl Mech

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The dynamic behavior of a three-layer elliptic cylindrical shell is studied and associated problems are formulated. The refined Timoshenko model is used. The numerical method used to solve the equations of motion is based on the integro-interpolation differencing method. The dynamic behavior of a three-layer elliptic cylindrical shell under a distributed nonstationary load is studied Keywords: three-layer elliptic cylindrical shell, numerical method, nonstationary vibrations Introduction. Three-layer cylindrical shells with elliptic cross-section are inhomogeneous throughout the thickness. Their inhomogeneity is due to layered structure and complex geometry. Studies concerned with static problems for noncircular cylindrical shells in various formulations based on nonconventional approaches are reviewed in [11]. Numerical results on the stress-strain state of laminated plates and shells are presented in [5]. The influence of geometrical parameters on the stress state of longitudinally corrugated cylindrical shells is examined in [9]. The stress-strain state of cylindrical shells with diagonal notches is analyzed in [10]. The influence of orthotropy on the stress-strain state in nonthin elliptic cylindrical shells is examined in [11]. The thermoelastoplastic deformation of noncircular cylindrical shells is studied in [12]. The free vibrations of three-layer elliptic cylindrical shells are addressed in [1].One of the specific features of the processes analyzed in dynamic problems for three-layer shells is their wave nature. Of interest and relevance is to study the forced nonaxisymmetric vibrations of three-layer elliptic cylindrical shells under nonstationary loads.We will derive the equations describing the nonaxisymmetric vibrations of a three-layer elliptic cylindrical shell based on a Timoshenko-type theory of multilayer shells with hypotheses applied to the whole layered structure and Reissner principle for dynamic processes [6]. The numerical method to be used to solve the equations of motion is based on the integro-interpolation differencing method for the spatial variables and an explicit-finite difference scheme for the time coordinate [7]. The dynamic deformation of a three-layer elliptic cylindrical shell under distributed internal pressure will be considered as a numerical example.1. Problem Formulation. Let us consider a three-layer elliptic cylindrical shell with mid-surface geometry defined as follows [2]:

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Forced vibrations of three-layer elliptic cylindrical shells under distributed loads

Meish

Mikhlyak

2010

Int Appl Mech

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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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Determining the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator

Galishin

Shevchenko

2011

Int Appl Mech

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A technique to determine the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator is developed. The technique is based on the theory of thin shells that takes into account transverse shear and torsional strains. Plastic equations that relate the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor are used as constitutive equations. The nonlinear scalar functions in the constitutive equations are found from base tests on tubular specimens under proportional loading for different stress modes. The boundary-value problem is solved by numerically integrating a system of ordinary differential equations Keywords: plasticity, stress state mode, shells of revolution Introduction. The stress-strain state (SSS) of shell structures determined with allowance for plastic strains was addressed in [1-3, 5, 9-12, etc.]. These publications use various constitutive equations based on a linear relationship between the mean stress (the first invariant of the stress tensor) and the mean strain (the first invariant of the strain tensor) and describing the deformation of shells with small elastoplastic strains disregarding the third invariant of the stress deviator. Constitutive equations relating the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor were proposed in [6,7]. The chosen measures of stresses and strains are energy-consistent. The equations are based on a nonlinear relationship between the first invariants of the stress and strain tensors. This relationship as well as that between the second invariants of the stress and strain deviators is determined from tests on tubular specimens under proportional loading for different stress modes. These studies were continued in [8].The purpose of the present paper is to develop a technique for determining the axisymmetric elastoplastic SSS of thin shells from kinematic and static equations that take into account the transverse-shear and torsional strains and the constitutive equations given in [6][7][8] and to validate the technique against test examples.1. Problem Formulation. Consider a thin shell of revolution made by joining elements with differently shaped meridian. The position of an arbitrary point of the shell is defined by curvilinear orthogonal coordinates s, j , V, where s ( ) s s s n 0 £ £ is the the meridional coordinate, j is the circumferential coordinate, and V is the distance from the point to the coordinate surface. Assume that Vk s and Vk j (here k s and k j are the principal curvatures of the coordinate surface) can be neglected compared with unity. We also neglect the normal stress s VV compared with the other normal stresses, i.e.,

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Thermoelastoplastic deformation of noncircular cylindrical shells

Cited by 3 publications

References 18 publications

Forced vibrations of three-layer elliptic cylindrical shells under distributed loads

Forced vibrations of three-layer elliptic cylindrical shells under distributed loads

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

Determining the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator

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