Axisymmetric Stress State Research Articles

ЦИЛИДРИЧЕСКАЯ ОБОЛОЧКА С ЗАПОЛНИТЕЛЕМ

In the article we will consider cylindrical shells, their definition and properties of cylindrical shells, as well as examples of their application. We will also consider methods for calculating cylindrical shells. A cylindrical shell is a geometric body formed by turning a rectangle around one of its sides. The result is a cylinder whose bases are two parallel and equal rectangles, and the side surface is a surface formed by rotating the rectangle around one of its sides. Cylindrical shells have two main elements – the base and the side surface. The bases of the cylindrical shell are parallel and equal rectangles, and the side surface is a surface formed by rotating the rectangle around one of its sides. A cylindrical shell with filling is considered. They are in an axisymmetric stress state. The case is considered when the filler has the shape of a hollow cylinder or an elastic cone. The results of calculations for each case are presented.

Experimental study on post-fire mechanical properties and fracture behavior of Q690 steel

Fire is one of the most dangerous disasters for high-strength steel structures. After a fire event, the deterioration of mechanical properties of high-strength steel together with the complex stress state generated by external loading may lead to an unexpected failure or collapse of high-strength steel structures. Therefore, it is very critical to evaluate the post-fire behavior of high-strength steel considering stress state effects. This study conducted a series of tests on the Q690 steel specimens after being exposed to high temperatures ranging from 200 to 1000°C with different cooling methods including air-cooling and water-cooling methods. The tested specimens including smooth round specimens, notched round specimens, and grooved plate specimens were designed to consider the failure mode in axisymmetric stress state and plane strain state. To validate the accuracy of the test results, the mechanical properties such as yield strength, ultimate strength, elastic modulus, and ultimate strain were first compared with other test results reported in the literature. The post-fire ultimate strength under different stress states was obtained directly from the test results. Then, the post-fire equivalent plastic strain at fracture of different specimens was obtained from test data together with the finite element analysis results and analyzed under different stress triaxialities and Lode angle parameters. According to this study, an increase in stress triaxiality can result in an increase in ultimate strength but a decrease in the equivalent plastic strain at fracture for Q690 steel after fire. When the exposed temperature was not higher than 800°C, the post-fire strength of the plane strain specimen (Grooved Plane Specimen) was higher than that of the axisymmetric tension specimen (Notched Round Specimen) under a similar stress triaxiality; however, for their post-fire equivalent plastic strain at fracture, the results were opposite.

Axisymmetric Stress State of an Adhesive Joint of Two Coaxial Pipes. An Improved Theory

Axisymmetric Stress State of an Adhesive Joint of Two Coaxial Pipes. An Improved Theory

ЭЛЕКТРОУПРУГИЕ СВОЙСТВА НАЧАЛЬНО-НАПРЯЖЕННОЙ ПЬЕЗОКЕРАМИКИ PZT-4 С ОДНОНАПРАВЛЕННЫМИ ТУННЕЛЬНЫМИ ПОРАМИ

Prediction results and numerical analysis of effect on effective transversal-isotropic electroelastic properties of porous piezoceramics of PZT-4 with unidirectional cylindrical (tunnel) pores of values of its initial axisymmetric stress state are presented. Effective constants have been identified: Young's and shear modules, Poisson's ratio in the transversal plane of isotropy and the "longitudinally/transverse" piezomodule of porous ceramics, which are most significantly influenced by the nature and values of its initial deformation in comparison with other practically "invariant" constants. It has been found that the "transversal" hydrostatic initial deformation of the porous ceramic has a greater effect on the effective constants of the porous ceramic than the same values of the "longitudinal" axial initial deformation.

Axisymmetric Stress State of Adhesive Joint of a Circular Patch with a Plate Weakened by a Circular Cut-out

Adhesive lap joints are widely used in modern structures. Known analytical mathematical models of the stress state of lap joints describe the joints of straight beams or cylindrical coaxial pipes. It is assumed that the stress state of these structures depends on only one coordinate. The study of the stress state of plates with defects, which are reinforced by patches, in most cases requires the use of at least two-dimensional mathematical models. In this work, it is shown that the axial symmetry of the plate, the cut-out, the patch, and the applied load makes it possible to reduce the problem to a one-dimensional problem in the polar coordinate system. An adhesive lap joint with circular symmetry is considered for the first time. The stress-strain state of the structure is described in an analytical form. Comparison of the results of calculating the stress state of the joint with the results of finite element modelling showed good adequacy of the proposed mathematical model.

A shear modified enhanced Gurson constitutive relation and implications for localization

The most widely used phenomenological constitutive relation to model ductile failure of structural metals at room temperature is based on the work of Gurson (1977) and has the advantage of having both a micromechanics basis and sufficient simplicity to enable complex engineering calculations to be carried out. The predictions of this framework have been most successful in circumstances where the stress triaxiality, the ratio of mean normal stress to Mises effective stress, is relatively large. For stress states with smaller values of stress triaxiality, the predictive capability has typically been much reduced. We present a shear modified enhanced Gurson constitutive relation that combines the shear modification of Nahshon and Hutchinson (2008) with the second porosity concept of Gologanu et al. (1994). This maintains both the connection with micromechanics and the computational simplicity. The predictive capability of the framework is illustrated by the good agreement with cell model calculations for localization of deformation over a wide range of stress states. Strain localization analyses with an initial porosity imperfection predict that for axisymmetric stress states with a superposed hydrostatic tension, the minimum critical strain for the onset of localization follows the onset of coalescence, as defined within the context of the constitutive model here, if the value of the stress triaxiality is sufficiently small. For an imposed shear stress state with a superposed hydrostatic tension, the minimum critical strain for the onset of localization is predicted to precede the onset of coalescence. A strong sensitivity of the critical strain for localization of deformation to the strain or stress range over which void nucleation occurs is also predicted, with void nucleation and localization of deformation essentially coinciding for sufficiently abrupt nucleation.

Deformation distortions of geometric form tube blanks in the process of drilling holes

The influence of deviations of the shape of the cross-sections of pipes from roundness on the axisymmetric stress state and taking this factor into account in theoretical studies and methods for solving problems of mandrel drilling in the elastic-plastic mode are considered. The features of the choice of tubular blanks, which help to reduce the unevenness of deformation and increase the accuracy of processing in the process of mandrel drilling, are revealed. Recommendations are given for eliminating the variance in wall thickness of blanks at the stage of their preparation for the mandrel operation.

Geometry of principal stress trajectories for piece‐wise linear yield criteria under axial symmetry

AbstractRigid‐ and elastic‐ plastic solids obeying quite a general yield criterion represented by linear functions of the principal stresses are considered. A general axisymmetric state of stress satisfying the hypothesis of Haar and von Karman is analyzed in quasi‐static flow. The circumferential velocity vanishes. A superimposed restriction on the yield criterion is that the system of stress equations is hyperbolic. The primary objective of this study is to select optimal coordinates and unknowns for deriving the integrable differential relations in their most convenient form for numerical treatments. It is shown that there exists a simple relation between the scale factors of the principal lines coordinate system (the coordinate curves of this coordinate system coincide with the trajectories of the principal stresses). Another simple relation exists between the scale factors and the principal stresses. The mapping between the principal lines and cylindrical coordinates is determined by a system of hyperbolic differential equations.

Axisymmetric Stressed State of Uniformly Layered Space with Periodic Systems of Internal Disc-Shaped Cracks and Inclusions

Using the Hankel integral transform, we construct discontinuous solutions for the problem of the axisymmetric stress state of a piecewise homogeneous, uniformly layered space, obtained by alternately connecting two heterogeneous layers of the same thickness. The space on the middle planes of the first heterogeneous layer contains a periodic system of circular disc-shaped parallel cracks, and on the middle planes of the second layer has a periodic system of circular disc-shaped parallel rigid inclusions. The determining system of equations is obtained in the form of a system of integral equations with kernels of the Weber --- Sonin type with respect to the crack extension and tangent contact stresses acting on the facial surfaces of rigid inclusions. With the help of rotation operators, the resulting determining system of equations is reduced to a system of integral equations of the second kind of Fredholm type. The equation solution is constructed by the method of mechanical quadratures. A numerical analysis was carried out and regularities were revealed in the variation of the intensity factors of rupture stresses, crack extension and contact stresses under the inclusions depending on the physical and mechanical and geometrical characteristics of the problem

Evaluation of additional confinement for three-dimensional geoinclusions under general stress state

Three-dimensional cellular geoinclusions (e.g., geocells, scrap tires) offer all-around confinement to the granular infill materials, thus improving their strength and stiffness. The accurate evaluation of extra confinement offered by these geoinclusions is essential for predicting their performance in the field. The existing models to evaluate the additional confinement are based on either a plane-strain or axisymmetric stress state. However, these geoinclusions are more likely to be subjected to the three-dimensional stresses in actual practice. This note proposes a semi-empirical model to evaluate the additional confinement provided by cellular geoinclusions under the three-dimensional stress state. The proposed model is successfully validated against the experimental data. A parametric study is conducted to investigate the influence of input parameters on additional confinement. Results reveal that the simplification of the three-dimensional stress state into axisymmetric or plane-strain condition has resulted in inaccurate and unreliable results. The extra confinement offered by the geoinclusion shows substantial variation along the intermediate and minor principal stress directions depending on the intermediate principal stress, infill soil, and geoinclusion properties. The magnitude of additional confinement increases with an increase in the geoinclusion modulus. The findings are crucial for accurate assessment of the in situ performance of three-dimensional cellular geoinclusions.

Bending of Axisymmetrically Loaded Thick Plates

We proposed a new theory of bending of axisymmetrically loaded plates in the form of thick rings or disks for the case where their deflections are not large and the stressed state is not described by the Kirchhoff–Love or Timoshenko hypotheses. For bending of this kind, we use two harmonic functions that describe the axisymmetric stressed state. After integration over the thickness of the plate, the momentsand transverse forces are expressed via two functions. The relationships of the theory of elasticity are exactly satisfied and a closed system of equations for the introduced functions is constructed without using any hypotheses about the geometric nature of deformation of the plate. A method for the solution of these equations is developed. We also present the solution of various problems of bending of the plates with holes.

Undrained lower bound solutions for end bearing capacity of shallow circular piles in non‐homogeneous and anisotropic clays

SUMMARYThe undrained bearing capacity of shallow circular piles in non‐homogeneous and anisotropic clay is investigated by the lower bound (LB) finite element limit analysis (FELA) under two‐dimensional (2D) axisymmetric condition using second‐order cone programming, and the new solution of the problem is presented. Modified from the isotropic von Mises yield criterion, a cross‐anisotropic undrained strength criterion of clays under the axisymmetric state of stress requiring three input shear strengths in triaxial compression, direct simple shear, and triaxial extension is employed in the 2D axisymmetric LB FELA. Parametric studies on the effects of pile embedment ratio, dimensionless strength gradient, anisotropic strength ratio, and pile roughness are investigated extensively, while the predicted failure mechanisms associated with these parameters are discussed and compared. Numerical results of undrained end bearing capacity of shallow circular piles are summarized in the form of design tables that are useful for design practice and represent a new contribution to the field of pile capacity considering the combined effects of undrained strength non‐homogeneity and anisotropy.

Validation of thick-walled cylinder analogy for modelling corrosion-induced concrete cover cracking

The main objective of this paper is to validate the use of the thick-walled cylinder (TWC) analogy to model corrosion-induced crack propagation in reinforced concrete (RC) structures. The TWC analogy neglects the effects of member geometry to create an idealized axisymmetric stress state. Although this simple approach has been used by many researchers, its suitability to various geometries and reinforcement configurations has not been comprehensively investigated previously. To achieve this goal, four different geometries are modeled using finite element analysis. The finite element models are two-dimensional, and they consist of plane membrane, quadrilateral elements under plane-stress conditions. Corrosion-induced expansion at the rebar/concrete interface is simulated using truss elements subjected to an equivalent thermal load. As a part of this validation, the effect of an adjacent corroding rebar and asymmetry of geometry were also investigated. The results are compared against the analytical solutions found using the TWC analogy. This study found that, in general, the use of the TWC analogy to model corrosion-induced cover cracking is appropriate; however, its limitations are also discussed.

Stress-fractional soil model with reduced elastic region

The original stress-fractional plasticity approach was developed for the axisymmetric stress state. However, it cannot accurately capture the stress-strain behaviour, e.g., quasi-steady state flow, temporary stress peak and state-dependent dilatancy of soils under different loading conditions. In this study, a modified fractional-order plasticity model for soils under the general stress state is developed using an alternative yielding surface with a reduced elastic region at the dry side of the critical state line. The proposed model is then validated against the results of a series of drained and undrained tests conducted on different soils, including sand, rockfill, ballast and clay, subjected to different loading paths. It is observed that the proposed model can reasonably simulate the key features, such as the hardening/softening, contraction/dilation, liquefaction, quasi-steady state flow and steady state flow behaviour, of different soils.

Resilient behavior of coarse granular materials in three-dimensional stress state

The traffic-induced cyclic stresses on the road base and subbase courses are usually in three-dimensional stress state, while so far most laboratory studies have focused on the deformation behavior of base and subbase layers in axisymmetric stress state. This study investigates the three-dimensional resilient behavior of coarse granular base and subbase materials based on a true triaxial apparatus. The factors of effective confining pressure, [Formula: see text], amplitude of cyclic shear stress, qampl, coefficient of cyclic intermediate principal stress, bcyc, and fines content are involved. Test results indicate that the increase of either [Formula: see text] or qamplleads to a nonlinear growth of resilient modulus with a decreasing growth rate. bcyc, which is proposed to represent the coupling of cyclic major and intermediate principal stresses, is found to have a promotion effect on the resilient stiffness, and the promotion effect tends to be enhanced by the increase of qampl. A modified model is established to predict the resilient modulus of coarse granular materials in three-dimensional stress state. There appears to exist a critical value of fines content, at which the variation trend of resilient modulus is changed from increasing to decreasing, and the critical fines content is related to both [Formula: see text] and bcyc.

Application of the Variational Method of Homogeneous Solutions for the Optimal Control of the Axisymmetric Thermoelastic State of a Cylinder

We propose a variational approach to the solution of the problem of optimization of a stationary axisymmetric thermal stressed state of a finite solid cylinder by controlling the distribution of volumetric heat sources. The proposed approach is based on the variational method of homogeneous solutions developed earlier for the solution of axisymmetric problems of the theory of elasticity for a cylinder. We study the influence of the height-to-radius ratio of the cylinder on the optimal values of the objective functional and the stressed state.

Elastic-plastic stress-strain state and strength of thick-walled pipe under the action of internal pressure

The task of calculating a plane axisymmetric elastic-plastic stress state of a thick-walled steel cylindrical pipe under the action of uniform internal pressure is considered. Thick-walled steel pipes are used in mechanical engineering, in heat engineering and heat-and-power engineering, in oil-producing and oil-refining industries, in the chemical industry. When loading in a pipe, two limiting states are realized: the elastic and plastic resistance of the material for two limiting values of pressure. For an intermediate pressure value, the cross-section of the pipe consists of two concentric annular zones – an inner elastic-plastic zone and an outer elastic zone. The exhaustion of the bearing capacity of the pipe occurs when the elastic-plastic deformation zone, spreading from the inner surface of the pipe, reaches its outer surface. Therefore, the presence of an elastic deformation zone at the outer surface of the pipe does not lead to the destruction of the structure. The aim of the work is to create an analytical technique for determining the calculated internal pressure, satisfying the strength condition under elastic-plastic deformation of steel cylindrical thick-walled pipes of different geometry, and leading to the most complete realization of the mechanical properties of the material. Recommendations on the effective search for the specified internal pressure are given.

The Development of the Method for the Calculation of the Shaping Force in the Production of Vehicle Wheel Rims

The objective of the paper is the development of the method for the calculation of the required force of shaping of a horizontal shape-forming machine used in the preproduction workflow. The design of a machine for horizontal shape-forming of wheel steel rims enables quick readjusting and transfer to another rim shape, which is significant under the conditions of frequent alteration of the product nomenclature. We developed the calculation method based on the assumptions of the deformations continuity, the constancy of the volume at plastic deformation and axisymmetric stressed state as well as the local application of the load. The research resulted in obtaining the analytical dependences for the calculation of the force of shaping of a wheel rim, taking into account the degree of deformation, the mechanical characteristics of the metal, hardening as well as the design and technological factors of the process. The topicality of the development consists in the necessity for the high-quality and the least expensive choice of the equipment of the required rated power when designing a new workflow. The experimental data, carried out both under the production and laboratory conditions, confirmed the adequacy of the theoretical formulas. It is determined by the assumptions and solutions obtained based on the mathematical apparatus of the technical theory of the shells of revolution. The clearness of the method will make it possible to algorithmize the calculation process quite easily and to use it as a subprogram in an applied complex of technical preproduction. It will also allow determining the shaping process most significant factors influencing the value of the force without the direct use of the calculation. We recommend to use it at enterprises producing vehicle steel wheels by the method of radial-rotation shaping.

ELASTOPLASTIC STRESS STATE OF CYLINDRICAL PIPE UNDER INTERNAL PRESSURE

The paper deals with calculating the plane axisymmetric elastic-plastic stress state of a heavy wall steel pipe under the uniform internal pressure. Two limiting states are applied to the pipe: elastic and plastic resistance of material for two values of the pressure limit. At the intermediate pressure, the cross-section of the pipe consists of two concentric annular zones, namely the inner the outer elastoplastic zones. Exhaustion of the load-bearing pipe capacity occurs when the elastoplastic deformation generated by the inner pipe surface reaches its outer surface. Therefore, the presence of the elastic deformation zone on external pipe surface does not lead to the structural failure.

Deformation mechanisms of idealised cermets under multi-axial loading

The response of idealised cermets comprising approximately 60% by volume steel spheres in a Sn/Pb solder matrix is investigated under a range of axisymmetric compressive stress states. Digital volume correlation (DVC) anal`ysis of X-ray micro-computed tomography scans (μ-CT), and the measured macroscopic stress-strain curves of the specimens revealed two deformation mechanisms. At low triaxialities the deformation is granular in nature, with dilation occurring within shear bands. Under higher imposed hydrostatic pressures, the deformation mechanism transitions to a more homogeneous incompressible mode. However, DVC analyses revealed that under all triaxialities there are regions with local dilatory and compaction responses, with the magnitude of dilation and the number of zones wherein dilation occurs decreasing with increasing triaxiality. Two numerical models are presented in order to clarify these mechanisms: (i) a periodic unit cell model comprising nearly rigid spherical particles in a porous metal matrix and (ii) a discrete element model comprising a large random aggregate of spheres connected by non-linear normal and tangential “springs”. The periodic unit cell model captured the measured stress-strain response with reasonable accuracy but under-predicted the observed dilation at the lower triaxialities, because the kinematic constraints imposed by the skeleton of rigid particles were not accurately accounted for in this model. By contrast, the discrete element model captured the kinematics and predicted both the overall levels of dilation and the simultaneous presence of both local compaction and dilatory regions with the specimens. However, the levels of dilation in this model are dependent on the assumed contact law between the spheres. Moreover, since the matrix is not explicitly included in the analysis, this model cannot be used to predict the stress-strain responses. These analyses have revealed that the complete constitutive response of cermets depends both on the kinematic constraints imposed by the particle aggregate skeleton, and the constraints imposed by the metal matrix filling the interstitial spaces in that skeleton.