Analysis of the plastic zone at the corner point of interface

Kipnis, L. A.; Полищук, Тетяна Вікторівна

doi:10.1007/s10778-009-0170-2

Cited by 8 publications

(5 citation statements)

References 14 publications

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“…Substituting functions (20) into the boundary conditions (16), (17) and interface conditions (18), (19) (boundary conditions at the inner edge for a fixed annular plate) leads to a system of linear homogeneous equations for A, B, C, and D: Let us determine the constants A 1 and A 2 . Using (14) and (15), we get ( ) (see (12)), the boundary pressure can be related to the radius of the plastic zone as follows:…”

Section: Deriving the Characteristic Equationmentioning

confidence: 99%

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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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Section: Deriving the Characteristic Equationmentioning

confidence: 99%

“…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.…”

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confidence: 99%

Development of instability in a rotating elastoplastic annular disk

Lila

Martynyuk

2012

Int Appl Mech

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“…Figures 1 and 2 show, by solid lines, the plastic zones (aka fracture process zones) for plane strain and plane stress, respectively, for tensile loads p p k The aspect ratio of the plastic zone is of the order of 0.7-0.8 for plane strain and 0.8-1.0 for plane stress, the stress state within the plastic zones being compound and inhomogeneous. This suggests that it is beyond reason to model plastic zones by strips with zero width or by cuts [12,13,17] with indefinite length and direction to which a simple load equal to the yield strength is applied. It is obvious that such a model has nothing to do with the real stress state.…”

Section: Tension Of a Body With A Crack Plane Problemmentioning

confidence: 99%

Discretization of the plane problem for a cracked body with nonlinear stress–strain diagram under tension

Khoroshun

2011

Int Appl Mech

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The plane problem for a cracked body with a piecewise-linear stress-strain diagram under tension is reduced by the Fourier transformation to a system of nonlinear algebraic equations. The system is numerically solved for plane strain and stress states of a perfect elastoplastic material to study plastic zones, stress and strain distributions, and displacements of crack faces Keywords: crack, discretization of a nonlinear problem, plastic zone, stress and strain distributions, crack opening displacement Introduction. One of the main divisions of fracture mechanics is the mechanics of cracks that studies the stress-strain state around a crack to establish criteria and patterns of its development. A rigorous approach to determining the limiting equilibrium state of a solid with a crack is known [3] to employ classical failure criteria for stresses or strains at the crack tip and allow for all the segments of the real stress-strain diagram. However, the difficulties related to the problem formulation and computation gave rise to alternative approaches based on additional assumptions on stress distribution and crack-tip fracture mechanism. Among them, the most popular are energy-, stress-, and strain-based failure criteria.The use of approximate approaches raises the question whether the simplifying assumptions and associated failure criteria adequately describe the real physical and mechanical processes at the crack tip and fit the rigorous mathematical formulation of the problem. This question can only be answered after a thorough analysis of the simplified failure criteria, experimental data, and the exact solution of the rigorously formulated problem.Griffith's energy criterion for brittle fracture [9], including its modification for quasibrittle fracture by Orowan and Irwin [11,19], is considered a basic failure criterion in crack mechanics. This criterion is based on the idea of balance between the elastic strain energy released as the crack grows and either the increment of the surface energy in the case of brittle fracture or the energy dissipated in the plastic zone at the crack tip in the case of quasibrittle fracture. However, this idea is incorrect [14]. Indeed, solving the nonlinear problem leads to infinite stresses at the crack tip, whereas the increment of surface energy and the work of plastic deformation are expressed in terms of finite stresses consistent with the real stress-strain diagram. This means that the stresses discontinue at the point separating the elastic and fracture ranges, despite what continuum mechanics tells us. Griffith's criterion will fail if we consider that the stresses in the range where the elastic strain energy is released are finite. Moreover, the increment of surface energy equal to the work done to separate two adjacent atomic layers is at least an order of magnitude less than the released elastic strain energy stored in at least ten atomic layers (concept of continuity), i.e., these quantities are incommensurable.The J-integral or the Cherepanov-Rice criterion [8] is a ...

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“…Modeling of fracture process zone at the tip of a crack reaching the non-smooth interface between different materials was performed by Kaminsky et al (2008). An analytical analysis of the plastic zone at a corner point of an interface was carried out by Kipnis and Polishchuk (2009).…”

Section: Introductionmentioning

confidence: 99%

Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding

Лобода

Lapusta

Sheveleva

2010

International Journal of Solids and Structures

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a b s t r a c tA plane problem for two identical piezoelectric semi-infinite spaces adhered by means of a thin isotropic interlayer is considered. It is assumed that a crack of a limited electric permeability occurs in the interlayer parallel to its faces. Combined electromechanical loading is prescribed at infinity. It is assumed that the interlayer is softer than the adherent materials. To avoid the singularities, which are typical for the Griffith crack model, two distinct zones -a zone of mechanical yielding and a zone of electrical saturation -of unknown lengths are introduced as crack continuations. These lengths can be essentially different, with the zone of mechanical yielding significantly longer or shorter than the zone of electrical saturation. Assuming that the interlayer thickness tends to zero, a constant normal stress is prescribed in the zone of mechanical yielding and a saturated electrical displacement is prescribed in the zone of electrical saturation. Outside of these zones, the semi-infinite spaces are assumed to be perfectly bonded. This formulation results in a linear fracture mechanics problem with unknown pre-fracture zone lengths. The problem, formulated mathematically by a system of two equations of linear relationship, is solved exactly. The unknown yield and saturated zones lengths are found from the conditions of finiteness of stress and electrical displacement at the ends of these zones for the both cases when the electrical saturated zone is longer and shorter than the zone of mechanical yielding. It is shown that the same equation as for the Griffith crack model can be used for the determination of the electrical displacement in the crack region. The main results of the paper are obtained in the form of simple analytical equations which are convenient for engineering applications. Some numerical illustrations in graphical and tabular form show dependencies of the pre-fracture zone lengths, the energy release rate, the mechanical displacement and electrical potential jumps on the electromechanical loading and the electrical permeability of the crack medium.

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Analysis of the plastic zone at the corner point of interface

Cited by 8 publications

References 14 publications

Development of instability in a rotating elastoplastic annular disk

Development of instability in a rotating elastoplastic annular disk

Discretization of the plane problem for a cracked body with nonlinear stress–strain diagram under tension

Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding

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