Formation of the plastic zone in an anisotropic body with a crack

Kurchakov, E. E.; Gavrilov, G. V.

doi:10.1007/s10778-009-0120-z

Cited by 6 publications

(7 citation statements)

References 11 publications

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“…Let us restrict the consideration of an isotropic body for which the components of the tensor F can be expressed in terms of two constants (r and s): The components of the tensor G are as follows [16]: Note that Eqs. (1.9) are identical with Kauderer's equations [15].…”

Section: Constitutive Equationsmentioning

confidence: 99%

Modeling the fracture process zone near a crack tip in a nonlinear elastic body

Kaminsky

Kurchakov

2011

Int Appl Mech

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The model of a fracture process zone near the tip of a mode I crack in a nonlinear elastic body is proposed. Using the numerical solution of an appropriate boundary-value problem, the effect of the fracture process zone on a crack opening displacement is examined Keywords: nonlinear elastic body, crack, fracture process zone 1. Introduction. As experiments [1,3,8,12] show, fracture occurs in a local area near a crack tip (fracture process zone) where extremely high stresses give rise to submicrocracks that grow and coalesce. The shape and structure of this zone depend on the material properties and loading conditions.The strains in the fracture process zone are very high (more than 50% in an elastoplastic body [4]). This is why many attempts to describe the deformation process in this zone using the small-strain deformation theory of plasticity led to physically incorrect results that disagree with experimental data [1].A great many theoretical studies of the stress-strain behavior near a crack tip in nonlinear bodies disregarding the fracture process zone were published in the 1960-70s [1,7,8]. Researchers presumed that by solving nonlinear boundary-value problems, it was possible to eliminate the singularity of the stresses and strains at the crack tip, which would be consistent with reality.However, the singularity was still there, but of different type. Thus, stresses and strains at the crack tip remained infinitely high.The lack of information on fracture mechanisms is compensated for by modeling the fracture process zone [1, 3, 10-13] based on experimental data [1, 8] on the localization of nonlinear deformation in narrow zones (slipbands) at the crack tip. This phenomenon is especially typical for the early stage of loading. Since the fracture process zone is usually located on the continuation of the crack and is much shorter than the crack, it is modeled, expanding upon the Leonov-Panasyuk model [6], by a slit with faces subject to certain stresses.The experiments [4] indicate that the fracture process zone occurs inside the zone of nonlinear deformation near the crack tip. We will study the effect of the fracture process zone on the opening displacement of a mode I crack. Use will be made of the model proposed in [3,11,12]. In this model, the length of the fracture process zone and, hence, the length of the slit remain constant with increase in the external load. What do change are the stresses applied to the faces of the slit. They are determined by solving a boundary-value problem and satisfying the continuity and limitation conditions for the stresses at the slit tip.We will analyze the role of the fracture process zone near a crack tip in a nonlinear elastic body in the case of a generalized plane stress state assuming small strains. The boundary-value problem will be formulated in terms of the displacement components. By solving the problem, we will ascertain how the fracture process zone influences the crack opening displacement.1. Preliminaries. To formulate the boundary-value problem, we need th...

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Section: Constitutive Equationsmentioning

confidence: 99%

Modeling the fracture process zone near a crack tip in a nonlinear elastic body

Kaminsky

Kurchakov

2011

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. The stress is maximum at the elastic-plastic boundary.…”

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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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“…Plastic zones in anisotropic bodies are still poorly understood. They were studied in few publications [16][17][18], where several boundary-value problems for plane strain and generalized plane stress state were solved numerically and the influence of anisotropy and crack length on the size and shape of plastic zones was analyzed.Note that elastoplastic fracture under loads acting along a crack is studied by nonclassical fracture mechanics because the classical Griffith-Irwin-Barenblatt and Leonov-Panasyuk-Dugdale approaches fail to describe such processes [11,17]. It is therefore of current importance to thoroughly study this anomalous case to complete the overall picture of elastoplastic fracture.…”

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Influence of tension along a crack on the plastic zone in an anisotropic body

2010

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The influence of longitudinal loading on the size and shape of the plastic zone near a crack in an anisotropic body is analyzed. A generalized plane stress state is considered. A relevant boundary-value problem is solved numerically to study the behavior of the main plastic zone at the crack tip, a new plastic zone above the crack, and an additional plastic zone on the lateral surface, which merge to form a single plastic zone Keywords: anisotropic body, crack, plastic zone, merging of two plastic zones Introduction. Various crack models are widely used in elastoplastic fracture mechanics [5,[8][9][10][11][12][13][14][15]. These models can be justified only if the size and shape of plastic zones near cracks are known. In this connection, it is extremely important to solve the relevant boundary-value problems. Many boundary-value problems for plane and antiplane strain as well as generalized plane stress state were solved (by analytic and numerical methods) in [3, 4, 6, etc.]. However, all of them deal with plastic zones in isotropic bodies. Plastic zones in anisotropic bodies are still poorly understood. They were studied in few publications [16][17][18], where several boundary-value problems for plane strain and generalized plane stress state were solved numerically and the influence of anisotropy and crack length on the size and shape of plastic zones was analyzed.Note that elastoplastic fracture under loads acting along a crack is studied by nonclassical fracture mechanics because the classical Griffith-Irwin-Barenblatt and Leonov-Panasyuk-Dugdale approaches fail to describe such processes [11,17]. It is therefore of current importance to thoroughly study this anomalous case to complete the overall picture of elastoplastic fracture.The present paper deals with a plastic zone near a crack in an anisotropic body. A generalized plane stress state is considered. The body is subject to tensile loads acting along and transversely to the crack. Strains are assumed small. The boundary-value problem is formulated in terms of the displacement components. By solving this problem numerically, we study the influence of longitudinal load on the size and shape of the plastic zone.1. Preliminaries. Assume that deformation involves no Pointing effect. Therefore, we can use tensor-linear constitutive equations to describe the deformation of the body.1.1. Tensor-Linear Constitutive Equations. The components of the stress tensor S and the components of the strain tensor D are related by the following tensor-linear equations [18]:

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Formation of the plastic zone in an anisotropic body with a crack

Cited by 6 publications

References 11 publications

Modeling the fracture process zone near a crack tip in a nonlinear elastic body

Modeling the fracture process zone near a crack tip in a nonlinear elastic body

Development of instability in a rotating elastoplastic annular disk

Influence of tension along a crack on the plastic zone in an anisotropic body

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