Conditions Of Plane Deformation Research Articles

Local slip of bodies caused by the inhomogeneous friction coefficient

We study the problem of contact interaction of two elastic isotropic bodies under the conditions of plane deformation with regard for slip caused by the local inhomogeneity of the friction coefficient under consecutive loading by normal and shear forces. By the method of complex potentials, this contact problem is reduced to a singular integral equation for the relative shift of the boundaries of the bodies in the region of slip whose solution is found in the analytic form. The influence of external loads on the relative shift of the boundaries of the bodies in this region, their length, and contact stresses is analyzed.

Investigation of the prefracture zone at the tip of a mode I crack reaching a nonsmooth interface of an elastic media

We perform calculation of the initial prefracture zone at the tip of a mode I crack that reaches a nonsmooth interface of two dissimilar elastic media at its corner point by the Wiener–Hopf method. The zone is modeled by a line of normal displacement fracture on a crack continuation. Expressions for the length of the prefracture zone and the potential energy accumulated in it are obtained. Their numerical values are compared with the corresponding values for the prefracture zone in a bonding material on the interface of the media, on the basis of which we draw a conclusion on the possible direction of development of the zone. The problem of an interface crack that lies on the interface of two dissimilar media has been extensively studied for a long time [14–17]. At the same time, the class of problems of a crack reaching the interface of a medium was given much less attention, though this problem is no less important for the mechanics of composite materials and adhesive and welded joints than the problem of an interface crack. The existing solutions were predominantly obtained for a crack that reaches a plane interface [1, 8, 18], whereas the problems of a crack with a tip at a corner point of the nonsmooth boundary of a joint, which is a stress concentrator with a power singularity, is of obvious interest [4]. Under loading, the stress concentration at the tip of a crack, at the corner point of the interface of a medium, leads to the formation of a prefracture zone in the vicinity of the crack. This zone affects substantially the stress-strain state in the region most dangerous from the viewpoint of further development of the crack and explains the necessity to investigate it in detail. In [3, 5–7], the initial stages of development of the prefracture zones localized on the interface of two media in the binding material were studied within the framework of models with lines of discontinuity of displacements. In particular, in [6], a calculation of the prefracture zone at the tip of a mode I crack that, under the conditions of plane deformation, reaches the nonsmooth interface of two dissimilar elastic homogeneous isotropic media at its corner point was performed under the assumption that the material that binds the parts of the body into a single whole is less crack-resistant than the material of the bound parts. However, if the condition of the smaller crack resistance of the binding material is discarded, another mechanism of formation of the prefracture zone is possible, namely, under a tensile normal stress, on a continuation of the crack, a prefracture zone can form in this direction. In what follows, we investigate the initial stage of the nucleation and development of such a prefracture zone.

Influence of hydrogen on the crack resistance of 10Kh15N27T3V2MR steel

In high ranges of pressure and temperature, hydrogen reduces the thresholds Δ Kth , cyclic fracture toughness Kfc , and short-term static crack resistance Kc of 10Kh15N27T3V2MR austenitic dispersion-hardened steel. Conditions of plane deformation and, correspondingly, conditions of determination of the characteristic K Ic are maintained in testing in the range 293 – 483 K of compact specimens 20 mm thick after their preliminary hydrogenation to a hydrogen content of 8 ppm. Low-cycle fatigue and sizes of the plastic zone under static crack resistance are most sensitive to hydrogen (and decrease by 80 and 70%, respectively, against those in helium), and the cyclic crack resistance parameters Δ K th and K fc (decrease by 25%) are least sensitive to hydrogen. The static crack resistance K c and the percentage reduction in the cross-sectional area ψ in short-term tension are reduced by 50%.

Elastoplastic compression of a rectangular body: An alternative approach

The elastoplastic compression of a body of rectangular cross section in a state of plane deformation is considered. In the proposed method, a system of nonlinear equations is solved by software minimizing the discrepancy of functions.

Prediction of influence parameters on the hot rolling process using finite element method and neural network

In the present investigation, a hot rolling process of AA5083 aluminum alloy is simulated using the finite element method. The temperature distribution in the roll and the slab, the stress, strain and strain rate fields, are extracted throughout a steady-state analysis of the process. The main hypotheses adopted in the formulation are: the thermo-viscoplastic behavior of the material expressed by Perzyna constitutive equation and rolling under plane-deformation conditions. The main variables that characterize the rolling process, such as the geometry of the slab, load, rolling speed, percentage of thickness reduction, initial thickness of the slab and friction coefficient, have been expressed in a parametric form giving a good flexibility to the model. The convergence of the results has been evaluated using experimental and theoretical data available in the literature. Since the FE simulation of the process is a time-consuming procedure, an artificial neural network (ANN) has been designed based on the back propagation method. The outputs of the FE simulation of the problem are used for training the network and then, the network is employed for prediction of the behavior of the slab during the hot rolling process.

Analysis of bed displacements under a periodic load

The problem of determining the displacements of a bed (linearly deformable and vertically inhomogeneous layer with a large thickness) under a periodic load is solved for conditions of plane deformation. The results obtained make it possible to calculate the maximum displacements of boundary points. Moreover, it is possible to replace the periodic load by a statically equivalent constant load, and the layer by a combined bed in which Winkler springs are added to the layer.

Interaction of plane harmonic waves with inclusions in the elastic space

We solve the problem of interaction of harmonic elastic waves with a thin elastic inclusion in the form of a strip in an unbounded body (matrix) under the conditions of plane deformation. In view of the small thickness of the inclusion, it is assumed that its bending and shear displacements coincide with the displacements of the corresponding points of its median plane. The displacements of the medium plane are found from the corresponding equations of the theory of plates. The method of solution is based on the representation of displacements in the form of singular solutions of the Lame equations with subsequent determination of the unknown jumps from singular integral equations. The indicated integral equations are solved numerically (by the collocation method). The relations for the approximate evaluation of the stress intensity factors at the ends of the inclusion are obtained.

Dynamical problem for an incompressible laminated cylinder with helical anisotropy. Part 2. Numerical analysis

For weakly compressible elastic helically orthotropic laminated cylinders loaded by internal impulsive pressure under conditions of plane deformation, we perform the numerical investigation of the character of convergence of one-dimensional wave solutions to the analytic relations for incompressible thick-walled shells obtained earlier by the author. An approximate engineering analytic procedure is proposed for the evaluation of the dynamic stress-strain state of inhomogeneous helically reinforced thick-walled cylindrical shells.

Pipe-defect assessment based on the limit analysis, failure-assessment diagram, and subcritical crack growth

We analyze the serviceability of a pipe containing a defect by the following three methods: limit analysis, failure-assessment diagram, and subcritical crack growth. The use of the proposed complex approach is explained by the fact that, despite the plasticity of pipe steels, they can suffer quasibrittle fracture (under the conditions of plane deformation) or fail as a result of plastic collapse. Moreover, the final fracture of a pipe with defect is preceded by the stage of stable crack growth. Therefore, in analyzing the serviceability of the pipe, one should also take into account the “leak-before-break” criterion. To take into account all these circumstances, it is necessary to use the indicated three methods, as in the case of evaluation of the serviceability of a pipe of API X52 steel containing a surface semielliptic notch. Defects of this sort reflect the influence of the surface gouges and dents responsible for the failures of pipes under the action of internal pressure.

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh( β)/ π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6 mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.

Изгиб и выпрямление листа в условиях плоской деформации

Изгиб и выпрямление листа в условиях плоской деформации

Averaging of the physicomechanical characteristics of piezocomposites (plane deformation)

The mechanical, electric, and piezoelectric characteristics of fibrous piezocomposites with doubly periodic system of fibers are averaged within the framework of a model of regular structures under the conditions of plane deformation. It is shown that, for the general case of stacking of fibers with arbitrary cross section, the problem is reduced to the determination of certain functionals specified by a system of linear algebraic and singular integral equations of the corresponding boundary-value problem of electroelasticity for the analyzed structure.

Effect of the Characteristics of Low-Temperature Plastic Deformation of Metals with bcc Lattice on the Stress-Strain State at the Tip of a Macrocrack

We present the results of application of a physical criterion of yield in the complex stress-strain state to the description of plastic deformation at the crack tip. Unlike the von Mises criterion, the proposed criterion enables one to take into account the characteristics of plastic deformation of bcc metals at low temperatures. We simulated the process of loading of a platelike specimen (400×1600 mm) containing a central crack under conditions of plane deformation. It is shown that the drop of shear resistance under the action of normal tensile stresses observed in the process of low-temperature plastic deformation of bcc metals and alloys strongly affects the parameters of the stress-strain state of the metal at the crack tip. Thus, it leads to the enlargement of the zone of local plastic strains, promotes an increase in the maximum level of plastic strains at the crack tip, and affects the relationship between the tangential and normal stresses in the plastic zone. It is demonstrated that the physical criterion of yield can be efficiently used for the numerical analysis of the stress-strain state at the crack tip in bcc metals and alloys at low temperatures by the finite-element method.

Manifestation of Hydrogen and Low-Temperature Brittleness in Near-Threshold Cyclic Crack Resistance of Materials

We experimentally show that the realization of conditions of plane deformation at the tip of a fatigue crack is not sufficient for guaranteeing the unique dependence of the crack growth rate on the range of the stress intensity factor, which is explained by the effect of crack closure. We describe advantages and disadvantages of the effective range of the stress intensity factor as a parameter that determines the mechanical conditions for the propagation of a fatigue crack. We analyze the phenomenon of positive influence of strengthening factors (a decrease in the temperature of testing and hydrogenation) on the cyclic crack resistance of materials in a low-amplitude range of loading determined with regard for the effect of crack closure. The decrease in the crack growth rate and the increase in fatigue thresholds are intensified as the level of loading decreases and the ductility of materials increases. Differences in the influence of strengthening factors in low- and high-amplitude ranges of loading are explained by different mechanisms of fracture controlled by the shearing strength and the tensile strength, respectively. We give several examples of the mechanical behavior of materials that show the inversion of the influence of hydrogen on the resistance to fracture: fatigue fracture of smooth steel specimens in gaseous hydrogen, high-temperature corrosion fatigue of preliminary hydrogenated titanium alloys, and the influence of hydrogenation on the wear resistance of structural steels in the process of friction and cavitation and on the parameters of cutting of a tool steel.

Stresses near Crack Tips on the Boundary of Two Media in the Presence of Plastic Strips

Under the conditions of plane deformation, we study stresses in a piecewise-homogeneous isotropic body near the tip of a mode I crack appearing at the angular point on the boundary of two media. It is assumed that plastic strips (modeled by plastic slip lines) are formed on the boundary of the media. To determine the stresses, we use the Mellin integral transformation and the Wiener–Hopf method. The angular point is a stress concentrator with power singularity and, therefore, immediately after the appearance of lateral plastic strips (zones), a new plastic zone begins to develop from this point. We study the dependence of the power of singularity of stresses on the angle made by the boundary and the elastic characteristics of the media.

Edge effects in composites reinforced by fibers with square cross section in the presence of a contact crack

We consider a two-component composite weakly reinforced by fibers with rectangular cross section and weakened by two contact cracks located symmetrically about one of the coordinate axes. The composite is under the conditions of plane deformation and subjected to the action of a uniform tensile load on its boundary. We study the influence of Poisson's ratios of the components of the composite on the character of the zones of edge effects. An approximate solution of the problem is found by the method of finite differences. We construct the zones of edge effects for the components of stresses depending on the values of Poisson's ratios of the components of the composite.

Influence of Hydrogen on the Formation of Fatigue Thresholds in Structural Steels

We analyze certain phenomena related to the influence of gaseous hydrogen and hydrogen dissolved in a metal on the near-threshold growth of fatigue cracks. The significant decrease in crack growth resistance due to the action of hydrogen as compared with that in vacuum is caused mainly by the adsorption decrease in strength due to adsorption. We established three factors of the ambiguous influence of hydrogen on the effective fatigue threshold ΔKth eff, for which a positive influence is replaced by a negative one, namely: the strength level, temperature of testing, and high-temperature degradation of the metal. The following fractographic peculiarities of the near-threshold growth of cracks in a degraded metal are revealed: the local tunneling along the front of a crack and the presence of fatigue grooves. We propose a mechanism of crack closure due to both roughness and the component of longitudinal shear at the tip of the crack. We analyze the scale effect of fatigue thresholds, determine the conditions for invariance of the parameter ΔKth eff under conditions of plane deformation, and established the dependence of ΔKth eff on the thickness of specimens in the case of tests of a hydrogenated degraded metal.

Solution of plane dynamic boundary-value problems of electromagnetothermoelasticity for cylindrical bodies

We give the mathematical statement of a two-dimensional dynamic problem of electromagnetothermoelasticity for cylindrical bodies and deduce the primary equations of electrodynamics and thermoelasticity for the complex problem. The primary equations of the two-dimensional dynamic problem of thermoelasticity in stresses are reduced to a system of hierarchically connected wave equations. The boundary-value problems are formulated for a long hollow cylinder and a cylindrical beam whose cross section has the shape of a circular sector. These bodies are in the state of plane deformation and subjected to the action of a nonstationary electromagnetic field on their outer surfaces. We propose a method for the solution of these boundary-value problems.

Tension of an elastoplastic disk. Part 2. Edge crack

Within the framework of the model of plasticity strips, we study the elastoplastic problem of fracture mechanics concerning the development of these strips near the tips of an edge crack in a circular disk. We assume that plastic deformations near the crack tip are localized along three plasticity strips (L 1,L 2,L 3) under the conditions of plane stressed state and along two plasticity strips (L 1,L 2) under the conditions of plane deformation. One of these strips (L 3) is modeled by a line of discontinuity of normal and tangential displacements and the remaining two strips (L 1,L 2) are modeled by the lines of discontinuity of tangential displacements. Their lengths and orientations are determined as a result of numerical solution of the problem by the method of singular integral equations. We also present the values of the crack-tip opening displacements.