Conditions Of Plane Deformation Research Articles

A FINITE ELEMENT ANALYSES OF THE EFFECT OF HYDROSTATIC PRESSURE IN A THIN ADHESIVE LAYER ON ITS FRACTURE UNDER MIXED-MODE LOADING

The problem of deformation of a sample, which is a composition of orthotropic bodies bound by an adhesive layer, realizing a mixed loading mode I + II for the layer material in the vicinity of a crack-like defect, is considered. A crack-like defect in a composite was considered both in the form of a mathematical section and a physical section, the thickness of which was considered as a linear parameter. A finite element solution with an elastic description of materials in a state of plane deformation is used. To find critical states, the values of specific elastic energy in the tip of a crack-like defect were analyzed. So, for the mathematical section, the values of energy were associated with stress intensity factors and for the physical section with the energy product in the form of the product of a linear parameter and the average specific free energy on the face of a dead-end finite element of the adhesive layer. Finding the characteristics averaged over the layer thickness was determined within the framework of the Neuber – Novozhilov approach. For a small finite value of the linear parameter the convergence of the energy product to the critical value of the specific elastic energy found for the crack model in the form of a mathematical cut is shown for equivalent loading of the sample with a physical cut and the linear parameter tends to zero. For an adhesive layer of finite thickness an energy fracture criterion is considered which takes into account the loosening of the material due to hydrostatic pressure. The loosening of the material was taken into account by the parameter which was determined from the solution of the problem of normal tension of the adhesive layer by the cantilevers of the two-cantilever beam taking into account the critical values of the specific elastic energy. A comparison is made of the calculated critical external load for the formulation of the problem using a layer of finite thickness and the classical representation of a crack-like defect in the form of a mathematical section and a layer of zero thickness under the appropriate failure criteria.

Research of orthotropic composites failure taking into account their structural stochasticity

The object of the study is the construction of the reliability assessing algorithm for the orthotropic composite plate, taking into account the stochasticity of its structure under the conditions of plane deformation. The plate consists of a matrix and reinforcement elements. The main orthotropic directions of the material coincide with the directions of the loading. The conducted studies are based on the failure criteria expressed through the components of macro stresses. The hypothesis of the weakest link is used, which for the case of the statistical theory of strength sounds like this: the ultimate (failure) loading for an orthotropic composite plate is equal to the ultimate loading for its weakest element. Defects-cracks are characterized by independent random variables – the half-length and the orientation angle between the defect line and the axis of orthotropy with a higher modulus of elasticity. The proposed model of orthotropic composite material corresponds to known experimental studies epoxy phenolic fiberglass on the cord glass fiber. The distribution probabilities density of defect orientation takes into account their predominant orientation in the direction of reinforcement. On the basis of the obtained composite failure loading integral probability distribution function, the construction and study of the dependence of the plate failure probability diagrams for different number of cracks, structural inhomogeneity and type of applied loading was carried out. Complex application of the composite materials fracture mechanics deterministic solution and the methods of probability theory and mathematical statistics allows for a more adequate assessment of the composite materials reliability, taking into account the stochasticity of their structure. The main content of this work is the construction and analysis of dependence of stochastically defective reinforced composite materials failure probability diagrams. The obtained results make it possible to evaluate the reliability of orthotropic stochastically defective materials under conditions of plane deformation. The algorithm of a compatible combination of defectiveness and randomness of the orthotropic composite material structure makes it possible to qualitatively investigate its failure under various types of applied loading.

On the Springback and Load in Three-Point Air Bending of the AW-2024 Aluminium Alloy Sheet with AW-1050A Aluminium Cladding.

This article presents the results of an analysis of the bending load characteristics and the springback phenomenon occurring during three-point bending of 1.0 and 2.0 mm thick AW-2024 aluminium alloy sheets with rolled AW-1050A cladding. A new proprietary equation was proposed for determining the bending angle as a function of deflection, which takes into account the influence of the tool radius and the sheet thickness. The experimentally determined springback and bending load characteristics were compared with the results of numerical modelling using different models: Model I, a 2D model for a plane deformation state, disregarding the material properties of the clad layers; Model II, a 2D model for a plane deformation state, taking into account the material properties of the cladding layers; Model III, a 3D shell model with the Huber-von Mises isotropic plasticity condition; Model IV, a 3D shell model with the Hill anisotropic plasticity condition; and Model V, a 3D shell model with the Barlat anisotropic plasticity condition. The effectiveness of these five tested FEM models in predicting the bending load and springback characteristics was demonstrated. Model II was the most effective in predicting bending load, while Model III was the most effective in predicting the amount of springback after bending.

Аналитическая модель квазихрупкого разрушения пластины с трещиной

The paper considers a rectangular plate with the edge crack of mode I of normal separation from the elastoplastic material with the ultimate strain. This class of materials includes, for example, the low-alloy steels used in structures operating at temperatures below the cold brittleness threshold. The strength of the plate was studied within the framework of the Neuber --- Novozhilov approach. The crack propagation criterion was formulated using the modified Leonov --- Panasyuk --- Dugdale model using an additional parameter, i.e., the plasticity zone diameter (pre-fracture zone width). Under conditions of small-scale yielding in the presence of the stress field singular feature in the vicinity of the crack tip, the two-parameter (dual) criterion for quasi-brittle fracture was formulated for mode I cracks in the elastoplastic material. The fracture dual criterion included deformation criterion at the crack tip, as well as the force criterion at the fictitious crack tip. The lengths of the original and fictitious cracks were differing by the length of the pre-fracture zone. Diagrams of the plate quasi-brittle fracture under conditions of plane deformation and plane stress were constructed. The parameters included in the proposed quasi-brittle fracture model were analyzed. It was proposed to select model parameters according to the approximation (σ--ε)-diagram of uniaxial tension and the KIc critical stress intensity factor

METHOD FOR CALCULATING TECHNOLOGICAL PARAMETERS FOR CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 2

The continuation of the methodology for calculating the technological parameters of the process of constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misalignment of the punch and die is outlined. The calculation of limiting cases of asymmetric extrusion, which, on the one hand, is reduced to a symmetrical extrusion, and, on the other hand, to a complete overlap of a small outflow gap, is considered. The above formulas make it possible to determine such important parameters of the stamping process as the total and specific deforming forces, as well as the heights of the walls formed as a result of extrusion.

METHOD FOR CALCULATING TECHNOLOGICAL PARAMETERS FOR CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 3

The end of the methodology for calculating the technological parameters of the process of constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misalignment of the punch and die is outlined. In this part of the methodology, the extrusion of a hardening material is considered, for which calculations are described in detail, taking into account hardening. The above formulas make it possible to determine such important parameters of the stamping process as the total and specific deforming forces, the heights of the walls formed as a result of extrusion, as well as the increase in yield stress.

METHOD FOR CALCULATING TECHNOLOGICAL PARAMETERS FOR CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 1

The beginning of the methodology for calculating the technological parameters of the process of constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misalignment of the punch and matrix is outlined. Described the definition of sections of the working stroke, which were free extrusion, partially constrained extrusion and completely constrained extrusion. The calculation of limiting cases of asymmetric extrusion, which, on the one hand, is reduced to a symmetrical extrusion, and, on the other hand, to a complete overlap of a small outflow gap, is considered. In the developed technique, extrusion of both non-hardening and hardening material is considered. In the latter case, the consideration of the hardening of the extruded material is described in detail. The above formulas make it possible to determine such important parameters of the stamping process as the total and specific deforming forces, as well as the heights of the walls formed as a result of extrusion.

DEFORMED STATE AND HARDENING OF THE WORKPIECE WITH CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 3

The continuation of the solution of the problem of determining the deformed state of the workpiece with constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned arrangement of the punch and matrix is presented. Formulas have been obtained that allow for any specific value of the punch’s working stroke to determine the current coordinates of material points and accumulated deformations in the plastic deformation region located under the punch near the extruded thick wall. Formulas are derived that allow taking into account the effect of hardening of a deformable material on stresses and force parameters of extrusion, as well as the geometric results of the achieved shape change.

DEFORMED STATE AND HARDENING OF THE WORKPIECE WITH CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 1

The beginning of solving the problem of determining the deformed state of the workpiece with constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned arrangement of the punch and matrix is stated. Formulas are obtained that allow for any specific value of the punch's working stroke to determine the current coordinates of material points and accumulated deformations in the area of plastic deformation located under the extruded thick wall of a rectangular bracket.

DEFORMED STATE AND HARDENING OF THE WORKPIECE WITH CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 2

The continuation of the solution of the problem of determining the deformed state of the workpiece with constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned arrangement of the punch and matrix is presented. Formulas are obtained that allow for any specific value of the punch’s working stroke to determine the current coordinates of material points and accumulated deformations in the plastic deformation region located under the punch near the extruded thick wall. Formulas are derived that allow taking into account the effect of hardening of a deformable material on stresses and force parameters of extrusion, as well as the geometric results of the achieved shape change.

THE KINEMATIC AND STRESS STATE OF THE WORKPIECE AT CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 4

On the basis of the complete system of equations of the theory of plastic flow, a solution to the problem of determining the kinematic and stress states of a workpiece with a completely constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned arrangement of the punch and matrix is presented. The determination of the power and geometric parameters of the process has been carried out. Formulas are obtained that are necessary to determine the main technological parameters of the process of extrusion of rectangular brackets with a relatively thin horizontal bridge.

Creep of an inhomogeneous polymeric cylindrical shell under heating

The article presents the calculation of a polymer thick-walled cylindrical shell, taking into account creep under the action of a uneven temperature field. The relevance of the work lies in the widespread intro-duction of polymer pipes in the repair and construction of pipelines for heating, sewerage, and water supply systems. The problem is solved in an axisymmetric formulation under plane deformation conditions. The calculation is based on the non-linear Maxwell-Gurevich equation, which is widely used in the calculations of polymer structures. In the problem under consideration, the cylindrical shell is in a flat de-formed state. It is also believed that temperature is a function of radius and time.

Numerical Simulation of the Failure Localization for Rock Massive Under Rock Pressure Conditions During Explosive Mining with Charges of Industrial Explosives

The complication of geomechanical conditions with an increase in the depth of field development requires improved calculations of drilling and blasting parameters based on studies of the patterns of changes in the stress-strain state and fragmentation of the blasted massif with an increase in rock pressure in various directions. This paper presents the results of studies of a section of a rock mass with a bare surface under conditions of flat deformation by the smoothed-particle hydrodynamics (SPH) method when blasting blasthole charges of sparing and crushing action. Regularities of change in time and space of the destruction zones of the massif and stresses with the formation of cracks as a result of the impact of an explosion in the presence of external pressure and in its absence are established. The results of calculations of the influence of physical and mechanical properties on the trigger effect of the destruction of the blasted rock are presented. To obtain preliminary estimates of the growth and branching of crack-like defects in real rocks under plane deformation conditions, numerical experiments were carried out. Diffraction patterns of fracture development on cracks filled with sand are obtained.

THE KINEMATIC AND STRESS STATE OF THE WORKPIECE AT CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 1

On the basis of the complete system of equations of the theory of plastic flow, the beginning of solving the problem of determining the kinematic and stress states of the workpiece with constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned position of the punch and matrix is presented. The determination of the flow rates and stresses in the area of plastic deformation located under the forming thick wall of the rectangular bracket has been carried out. Formulas are obtained that are necessary to determine the main technological parameters of the process of extrusion of rectangular brackets with a relatively thin horizontal bridge.

THE KINEMATIC AND STRESS STATE OF THE WORKPIECE AT CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 3

On the basis of the complete system of equations of the theory of plastic flow, the beginning of solving the problem of determining the kinematic and stress states of the workpiece with constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned position of the punch and matrix is presented. The determination of the flow rates and stresses in the area of plastic deformation located under the forming thick wall of the rectangular bracket has been carried out. Formulas are obtained that are necessary to determine the main technological parameters of the process of extrusion of rectangular brackets with a relatively thin horizontal bridge.

THE KINEMATIC AND STRESS STATE OF THE WORKPIECE AT CONSTRAINED EXTRUSION OF RECTANGULAR BRACKETS. Part 2

On the basis of the complete system of equations of the theory of plastic flow, the beginning of solving the problem of determining the kinematic and stress states of the workpiece with constrained extrusion of rectangular brackets under conditions of plane deformation in the general case of misaligned position of the punch and matrix is presented. The determination of the flow rates and stresses in the area of plastic deformation located under the forming thick wall of the rectangular bracket has been carried out. Formulas are obtained that are necessary to determine the main technological parameters of the process of extrusion of rectangular brackets with a relatively thin horizontal bridge.

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.

System of cracks under the impact of plane elastic waves

The problem of determining the two-dimensional diffraction field and the stress state arising from the interaction of elastic waves with a system of arbitrarily located cracks is solved. Cracks are located in an unbounded elastic medium which is in a plane deformation state. The method of solution is based on the use of discontinuous solutions of the equations of motion of elastic media and the reduction of the original problem to a system of singular integro-differential equations. The obtained systems are solved numerically the mechanical quadrature method.

Моделирование механических и физических свойств пучка углеродных нанотрубок при поперечном сжатии с использованием цепной модели с редуцированным числом степеней свободы

The paper studies a bundle of oriented carbon nanotubes (CNTs) under the transverse loading under the plane deformation conditions within the framework of a molecular dynamics model with a reduced number of degrees of freedom. The model takes into account CNT wall stretching and bending, as well as van der Waals interactions. Each CNT is represented by a ring of atoms with two degrees of freedom in the plane of the ring. The discrete nature of the model allows describing the large curvature of the CNT wall and the destruction of CNTs at very high pressure. CNT crystal equilibrium structures are obtained under the strain-controlled biaxial loading. Separate CNTs of a sufficiently large diameter have two equilibrium states: with a round and collapsed cross section. Small-diameter CNTs in the free state can only have a circular cross section. The study identified the presence of two phase transitions observed during biaxial compression of a CNT bundle. The first transformation similar to phase transition of the second order leads to ellipticization of CNT cross sections. As a result of the second transition of the first order, bundled CNTs appear in the beam, the proportion of which gradually increases with the increase in compressive strain. The authors calculated beam elasticity constants such as Young’s moduli, shear modulus, and Poisson’s ratios. The study shows that one of the equilibrium structures (with elliptical CNT cross sections) has the property of a partial auxetic, that is, it has a negative Poisson’s ratio under uniaxial loading in a certain direction. The proposed chain model can be effectively applied to analyze physical and mechanical properties of bundles of single-walled or multi-walled CNTs under the plane deformation conditions, and after simple modifications, it can be used to similar structures made of other two-dimensional nanomaterials.

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.