Conditions Of Plane Deformation Research Articles

PERIODIC REINFORCEMENT OF TWO PRE-STRESSED STRIPS BY FINISHED SUPPORTING ELEMENTS

The qualitative and quantitative influence of initial (residual) stresses on the law of distribution of contact characteristics at interaction of elastic finished overlays (stringers), at their placement, with two prestressed strips which are clamped by one edge. In general, the research was carried out for the theory of great initial (ultimate) and two variants of the theory of small initial deformations within the framework of linearized theory of elasticity with the elastic potential having arbitrary structure for compressible and incompressible bodies. Fourier integral methods, methods for solving harmonic differential equations, singular integral-differential equations and numerical methods were used too. It is assumed that in the contact area the initial stress state, with a certain degree of accuracy, is considered homogeneous. We make the assumption that the elastic band with initial (residual) stresses is in the conditions of plane deformation. Also, for an elastic lining loaded with both vertical and horizontal forces, the generally accepted model of beam bending in combination with the model of uniaxial stress-strain state of the elastic lining is valid. A singular integral-differential equation with the Gilbert kernel was derived, which allows us to solve this problem. We find the analytical solution of the equation in the form of series from the Jacobi function. Numerical studies were performed for materials with elastic potentials of harmonic type (compressible bodies) and elastic potentials of Bartenev-Khazanovich and Treloar (incompressible bodies). Numerical studies have been performed for materials with elastic potentials of harmonic type (compressible bodies) and elastic potentials of Bartenev-Khazanovich and Treloir (incompressible bodies). We have considered the case when all periodically placed pads that support elastic bands with the initial stresses were loaded with tangential force. Analysis of numerical results shows a significant effect of initial (residual) stresses on the distribution of contact characteristics of periodically reinforced strips with thin reinforcing elements. The results obtained can be used for engineering calculations for the strength and durability of structures, taking into account the initial stresses for a wide range of structural materials.

The Effect of Distance between Sheet Pile and Foundation on Bearing Capacity of Foundation

In this study, the presence of a foundation near the sheet pile wall built on sand soil was simulated in the laboratory, similar to the field conditions. For this purpose, laboratory tests of sheet pile wall (SPW) (defo) and bearing capacity (qult) of the foundation were carried out by applying vertical loads to the steel model foundation. Besides, laboratory tests were represented by the Mohr‐Coulomb material model under 3D plane deformation conditions, and numerical analyses were performed. These studies were performed by varying the distance among the SPW, the foundation (L), and the penetration depth of SPW (Hp). The results demonstrated that L had a greater effect on qult and δmax than the Hp. When L was kept constant and Hp was increased, it was seen that qult increased and δmax decreased. Moreover, the raise in L has resulted in an increment in the qult and δmax decreased. In addition, a good agreement between numerical analysis and experimental test was observed.

Power Mode of High-Speed Combined Extrusion of Flat Bimetallic Road Milling Picks

The paper presents a mathematical model developed for calculating the force effect on the punch in the process of high-speed combined hot extrusion of bimetallic road milling picks under plane deformation conditions. To solve the problem, the process is divided into two phases: acceleration phase and braking phase, which consists of two stages. A distinctive feature of the acceleration phase is that it allows the analysis of reverse extrusion, in the process of which the metal flows in the opposite direction to the punch stroke. A method for calculating the force acting on the punch at each phase of the process of plastic flow of a bimetallic workpiece into a matrix cavity with three deformation zones is presented in the paper. While solving the problem in a quasi-static formulation and proceeding from the conditions of the minimum power of internal forces, equations have been obtained for calculating the optimal field parameters aopt, bopt, gopt, depending on the elongation coefficients l and the friction coefficient m. The equations obtained within the framework of the developed model are quite correct, since they allow determining the minimum force acting on the punch. The considered calculation model and equations can be used in the development of industrial technology for high-speed combined hot extrusion of flat-step bimetallic road milling picks.

Research of stamping of unequal channel bars. Part 3. Force parameters and shape change of the billet when extruding channels. 1. Kinematic and stress states of the billet

On the basis of the complete system of equations of the theory of plastic flow, the kinematic and stress states of the billet are determined when the channels are extruded under conditions of plane deformation of the misaligned position of the punch and the matrix. Keywords: die forging, extrusion, misaligned position, punch, matrix, plane deformation, plastic flow rates, stresses. mt13@bmstu.ru

ПОЛЗУЧЕСТЬ НЕОДНОРОДНОЙ ПОЛИМЕРНОЙ ЦИЛИНДРИЧЕСКОЙ ОБОЛОЧКИ ПРИ НАГРЕВЕ

Abstract. The article deals with the problem of creep of a polymer thick-walled cylindrical shell under the action of an uneven temperature field. The calculation is based on the nonlinear Maxwell – Gurevich equation, which is widely used in the calculations of polymer structures. To solve a quasi-stationary nonlinear problem, a "layer-by-layer" method is used, where a layer means a certain time of the process. Such methods for solving creep problems are also called step, sequential loading, incremental, etc. The problem is solved in an axisymmetric formulation under conditions of plane deformation. The use of the Maxwell – Gurevich differential equation allows solving rheology problems when exposed to a timevarying temperature field. The change in the stress state with time and its significant difference from the elastic solution are shown.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ ИДЕАЛЬНО УПРУГОПЛАСТИЧЕСКОЙ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Abstract. The article considers the construction of differential equations of equilibrium in displacements for plane deformation of elastic perfectly plastic regarding shear deformations continuous medium and nonlinearly elastic continuous medium with respect to volumetric deformations with bilinear approximation of the closing equations, both regarding and regardless geometrical nonlinearity in a cylindrical coordinate system. Nonlinear diagrams of volumetric and shear deformation are approximated by bilinear functions. Proceeding from the assumption of independence, generally speaking, of volume and shear deformation from each other, five main cases of physical dependencies are considered, depending on the relative position of the break points of bilinear diagrams of volume and shear deformation. The construction of bilinear physical dependencies is based on the calculation of the secant moduli of volumetric and shear deformation. In this case, in the first section of the diagrams, the secant modulus of both volumetric and shear deformation is constant, while in the second section of the diagrams, the secant modulus of volumetric deformation is a function of volumetric deformation, and the secant shear modulus is a function of the intensity of shear deformations. Substituting the corresponding bilinear physical relations into the differential equations of equilibrium of a continuous medium, written both regardless and regarding geometrical nonlinearity, the resolving differential equations of equilibrium in displacements for plane deformation in a cylindrical coordinate system are received. The received differential equations of equilibrium in displacements in cylindrical coordinates can be applied in determining the stress-strain state of elastic perfectly plastic with respect to shear deformations continuous medium and nonlinearly elastic with respect to volumetric deformations continuous medium under conditions of plane deformation, both regarding and regardless geometrical nonlinearity, physical relations for which are approximated by bilinear functions.

Optimization of the geometry of specimens with transversal groove for materials with initial anisotropy

The goal of the study is determination of the optimal geometric dimensions of rectangular anisotropic specimens with a transverse groove when tested for uniaxial tension under conditions close to plane deformation. A test scheme in which the shear fracture mechanism is implemented in the center of the groove under conditions of plane deformation is presented. The optimal geometric dimensions of the specimen (namely, the depth, width and length of the transversal groove) were determined proceeding from numerical simulation of the experiment in the ABAQUS software package in the Explicit mode. It is shown that the initial anisotropy of the material significantly affects the deformed state in the groove. The larger the ratio of the Lankford coefficients along the rolling direction and across this direction, the closer the deformed state in the groove to the plane deformation. It is also shown that change in the geometry of the specimen with a transverse groove can provide better results in implementation of the plane deformation in the region of specimen fracture. The value of the parameter of the deformed state approaches zero when the thickness and width of the groove decrease, as well as when the width of the specimen increases. It is shown that the possibility of forming plane deformation in the groove due to the choice of specimen geometry is limited from above for any initial anisotropy. A moment of «saturation» is observed, at which the ratio of principal strains in the center of the groove does not go any further to zero despite further change in the geometry. A table of geometric dimensions of rectangular specimens with a transverse groove recommended for tensile testing under conditions of plane deformation for all types of anisotropic and isotropic sheet materials is presented.

ВСЕСТОРОННЕЕ ИССЛЕДОВАНИЕ ВЫДАВЛИВАНИЯ П-ОБРАЗНЫХ КРОНШТЕЙНОВ. СООБЩЕНИЕ 6. КИНЕМАТИЧЕСКОЕ И НАПРЯЖЕННОЕ СОСТОЯНИЯ ЗАГОТОВКИ ПРИ СТЕСНЕННОМ ВЫДАВЛИВАНИИ. Ч. 2

On the basis of the complete equation system of the plastic flow theory, the solution continuation of the determination problem of the kinematic and stressed states of a blanket during restricted extrusion of П-shaped brackets under conditions of plane deformation in the general case of a nonaligned placement of the punch and the die is stated. The determination of flow velocities and stresses in the area of plastic deformation, located under the punch end near the formed thick wall of the bracket has been carried out. The formulae were obtained, which are required for identification of the main parameters of the extrusion process of П-shaped products with a relatively thin horizontal bridge.

Calculation of an inhomogeneous polymer thick-walled cylindrical shell taking into account creep under the action of temperature load

The article presents the calculation of a polymer thick-walled cylindrical shell taking into account creep under the action of an uneven temperature field. The relevance of the work lies in the widespread introduction of polymer pipes in the repair and construction of heating pipelines, sewerage systems, water supply systems. The problem is solved in an axisymmetric setting under conditions of plane deformation. The calculation is based on the nonlinear Maxwell - Gurevich equation, which is widely used in the calculations of polymer structures.

ВСЕСТОРОННЕЕ ИССЛЕДОВАНИЕ ВЫДАВЛИВАНИЯ П-ОБРАЗНЫХ КРОНШТЕЙНОВ. СООБЩЕНИЕ 7. ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ И УПРОЧНЕНИЕ ЗАГОТОВКИ ПРИ СТЕСНЕННОМ ВЫДАВЛИВАНИИ. Ч. 1

The solution beginning of the determination problem of the strained state of a blanket during restricted extrusion of П-shaped brackets under the conditions of plane deformation in the general case of a nonaligned placement of the punch and the die is stated. The formulae were obtained, which allow one to determine current coordinates of material points and cumulative strain in the area of plastic strain situated under the extruded thick wall for any specific value of the working stroke of a punch.

ВСЕСТОРОННЕЕ ИССЛЕДОВАНИЕ ВЫДАВЛИВАНИЯ П-ОБРАЗНЫХ КРОНШТЕЙНОВ. СООБЩЕНИЕ 6. КИНЕМАТИЧЕСКОЕ И НАПРЯЖЕННОЕ СОСТОЯНИЯ ЗАГОТОВКИ ПРИ СТЕСНЕННОМ ВЫДАВЛИВАНИИ. Ч. 3

On the basis of the complete equation system of the plastic flow theory, the solution continuation of the determination problem of the kinematic and stressed states of a blanket during partially restricted extrusion of П-shaped brackets under the conditions of plane deformation in the general case of a nonaligned placement of the punch and the die is stated. The determination of the force and geometric parameter of the process was carried out. The formulae were obtained, which are required for identification of the main technological parameters of the extrusion process of П-shaped products with a relatively thin horizontal bridge.

ВСЕСТОРОННЕЕ ИССЛЕДОВАНИЕ ВЫДАВЛИВАНИЯ П-ОБРАЗНЫХ КРОНШТЕЙНОВ. СООБЩЕНИЕ 6. КИНЕМАТИЧЕСКОЕ И НАПРЯЖЕННОЕ СОСТОЯНИЯ ЗАГОТОВКИ ПРИ СТЕСНЕННОМ ВЫДАВЛИВАНИИ. Ч. 1

On the basis of the complete equation system of the plastic flow theory, the solution origination of the determination problem of the kinematic and stressed states of a blanket during restricted extrusion of П-shaped brackets under conditions of plane deformation in the general case of a nonaligned placement of a punch and a die is stated. The determination of flow velocities and stresses in the area of plastic deformation, located under a formed thick wall of the bracket has been carried out. The formulae were obtained, which are required for identification of the main parameters of the extrusion process of П-shaped products with a relatively thin horizontal bridge.

ВСЕСТОРОННЕЕ ИССЛЕДОВАНИЕ ВЫДАВЛИВАНИЯ П-ОБРАЗНЫХ КРОНШТЕЙНОВ. СООБЩЕНИЕ 6. КИНЕМАТИЧЕСКОЕ И НАПРЯЖЕННОЕ СОСТОЯНИЯ ЗАГОТОВКИ ПРИ СТЕСНЕННОМ ВЫДАВЛИВАНИИ. Ч. 4

On the basis of the complete equation system of the plastic flow theory, the solution of the determination problem of the kinematic and stressed states of a blanket during completely restricted extrusion of П-shaped brackets under the conditions of plane deformation in the general case of a nonaligned placement of the punch and the die is stated. The determination of the force and geometric parameter of the process was carried out. The formulae were obtained, which are required for determination of the main technological parameters of the extrusion process of П-shaped products with a relatively thin horizontal bridge.

Об одном варианте расчета напряженного состояния пласта, отрабатываемого очистной выработкой

An elastoplastic problem is formulated in which the boundary part of the seam in front of the face, as well as the part of the collapsed rock layer behind the face, are in the limit stress state. In a rock massif along the coal seam at its full capacity, the face of a rectangular section moves to the right. The rocks in the vicinity of the working are under the conditions of plane deformation. The characteristics of the of the coal seam strength are significantly inferior to the characteristics of the strength of the rocks of the enclosing massif but exceed the characteristics at the contact of the seam with the rest of the massif. The edge part of the coal seam in front of the working is in the limit stress state. Behind the working, a layer of a massif of rock that collapsed from above is formed after the coal extraction. The collapse of rocks starts from the roof of the working and extends into the depth of the massif at some angle to the horizon. Depending on the mining and geological conditions, this angle varies between 40–50°. The maximum thickness of the collapsed rock layer is determined by the conditions of cleavage, at which the upper layers rely on the collapsed rocks, preventing their further significant displacement.  The elastic-plastic problem of the stress state of the boundary part of the seam mined by the working is reduced to the second boundary problem of the elasticity theory, which is solved by the method of boundary integral equations. The parameters of the bearing pressure in the boundary part of the seam (in front of the face) slightly vary with the growth of the face span. The parameters of the limit stress zone in the collapsed layer behind the mine working in the entire range of variation of its span are an order of magnitude larger than the dimensions of the bearing pressure zone in the boundary part of the mined seam in front of the face. The maximum stresses in the collapsed rock layer are on average 1.5 times less than the maximum stresses in the boundary part of the mined seam.

Method of Determining a Thermomechanical Diagram for Stressed Cylinder Connections upon Their Plane Deformation

In this paper, we propose a new method for determining the thermomechanical diagram (TMD) when testing stressed cylinder connections under plane deformation conditions. The application of the TMD method is based on the identity of the specification and restoration of deformation on the samples and in the implementation of shape memory by a force element in the structure. Therefore, such a problem statement is necessary for calculating the contact load when forming pipe connections by couplings with the shape memory effect.

Improvement of the optical method for researching the contact stresses during the forging

Dobrov I. V., Semichev A. V. Improvement of the optical method for researching the contact stresses during the forging. Material working by pressure. 2020. № 1 (50). Р. 24-27.
 The analysis of the results of experimental resarch of contact normal stresses during the forging of a symmetrical workpiece under conditions of plane deformation performed by the tensometric method is carried out. It is shown that when comparing the combined graphs of the distribution of contact normal stresses along the symmetry axis of the workpiece, depending on the position of the considered section of the workpiece relative to the length (diameter) of the contact surface, presented in the technical literature, it is necessary to take into account the real change in the dimensions of the contact surface during forging of the initial workpiece with different degrees of deformation. An optical method for research of the distribution of contact normal stresses during forging under conditions of plane deformation of a symmetric workpiece made of a material with high plastic properties, which during forging, flows into holes of equal diameter, located along the axis of symmetry at an equal distance from each other in the diaphanous puncheon of new design. The results of experimental expand the physical understanding of the nature of the redistribution of normal contact stresses during forging and qualitatively coincide with the results of load cell measurements of normal contact stresses in real conditions presented in the scientific and technical literature.

Calculating the parameters of the induced hydraulic fracture that moves in hard rock roof of a seam in the vicinity of the installation chamber

The results of solving the task for tracing the movement of a hydraulic fracture in the top rocks situated in the vicinity of a stope are introduced in the paper. The massif is loaded with equally component gravity field of stresses and is in the state of plane deformation. The bedrocks and top rocks (roof rocks) are rather tough and deform resiliently and when reaching the limit state fall as brittle materials. The task is solved in the framework of the model of the geo-mechanical state of the rock massif with the adjacent in-seam working. It is based on fundamental principles of the deformable solids mechanics and fulfilled applying the boundary element method and basic classical concepts about the fracture state, its stable or unstable growth in a brittle material. According to the results of the experiment, the trajectories of the fracture growth for a range of coordinate values, its initial position and the angle of gradient initial towards the horizon are built. The graphs of critical pressure changes while the growing of a fracture are introduced.

The problem of pore fracture growth in a marginal part of a gas-bearing coal seam

The model of geomechanic state of the coal massif inclosing in-seam working driven along a gas-bearing seam is introduced in the paper. The model is built on the basis of the fundamental concepts of deformable body geomechanics and elastic fracture mechanics. It suggests that the massif is in the state of plane deformation loaded with a stress gravity field. It is assumed that strength characteristics of a coal seam are less than the strength characteristics of the inclosing seam rocks but higher than the strength characteristics along their contact. Coulomb-Mohr and Mohr-Kuznetsov strength criteria are taken as the condition for appearing of inelastic deformation. In this connection, the problem of stress-deformable state of a coal rock massif appears to be an elasticoplastic one. The pore gas pressure changes deep down the seam according to the hyperbolic tangent function. The model assumes that when setting the boundary condition on the seam contact with the side rocks the microfractures containing pore methane are oriented in the direction of a maximum principal stress.

Stress-Strain State of a Plastic Layer under Compression by Two Rigid Parallel Rough Plates

The stress-strain state of a plastic layer under compression by two rigid parallel rough plates under conditions of plane deformation is investigated. The hypothesis of flat sections is used. Tangential stresses are found in the part of the layer under load, as an approximate solution of the system of equations of the stress-strain state of the compressible layer. The solution is obtained in analytical form. On this basis the velocities of the points of the layer are calculated in the analytic form. The size and shape of the free surface are calculated depending on the thickness of the layer. The coefficients of this dependence are determined by recurrent relations. On this basis, an algorithm for finding the shape of a free surface is found. The computational experiments are provided and the results are compared with results obtained with FEM and ANSYS.

АНАЛИЗ КРИТИЧЕСКОГО СОСТОЯНИЯ ДИСКРЕТНО-НЕОДНОРОДНОЙ ПОЛОСЫ С НАКЛОННОЙ КОНТАКТНОЙ ГРАНИЦЕЙ И НАРУЖНЫМ МАКРОДЕФЕКТОМ В БОЛЕЕ ПРОЧНОЙ ЧАСТИ

The article considers the stress state of a joint in the form of an elastic-plastic strip with rectilinear parallel edges under conditions of plane deformation. The strip consists of two parts of different strengths made of homogeneous isotropic materials, separated by a straight line which is forming an arbitrary angle with the edge of the strip. The joint is subjected to tensile stress. In its stronger part, there is a surface macrodefect in the form of an external incision or recess. The purpose of the article is to study the critical state of the joint using various geometrical and mechanical parameters and, on this basis, calculate the critical tensile force. The method is based on the study of characteristic fields (slip lines) in the plastic zone, taking into account the presence of a voltage break in the more durable part. A complete analysis of variations of the patterns of characteristic fields leading to different stress diagrams over the net cross section is given, depending on the relative position of the defect and the contact boundary, the size of the defect, the angle of inclination of the contact boundary, and the coefficient of mechanical heterogeneity of the joint. An algorithm has been developed to calculate the critical tensile force in the general case. The average critical stresses in the most characteristic cases have been calculated.