The deformation model of the DCB-sample with elastoplastic properties

The loading of a strip with a crack-like defect according to mode I is considered. In contrast to the classical representation of a crack in the form of a mathematical section, the proposed model defines a crack as a physical cut with a characteristic linear size. The mental continuation of a physical cut in a solid forms an interaction layer (IL). It is important that the stress-strain state of the layer at a finite value of the linear parameter does not introduce a singularity into the crack model. The process of elastoplastic deformation with a constant layer length is considered. We obtained a simplified analytical solution to the problem of deformation of two elastic bodies connected by a thin layer with elastoplastic properties. The dependence of the displacement and stress fields on the length and thickness of the interaction layer has been found. It is shown that, under the classical plasticity condition, the range of variation of the external load leading to a purely elastic behavior is possible only for a finite layer thickness. As the layer thickness tends to zero, as in the Dugdale model, the plasticity region is formed at an arbitrarily small external load. For small layer thicknesses, a local plasticity criterion is proposed, by using which it is possible to distinguish the intervals of the external load variations associated with elastic and plastic deformations. The local plasticity condition, determined by the critical value of the energy product, makes it possible to reflect the stage of elastic deformation at an arbitrarily small finite thickness of the interaction layer. An asymptotic dependence of the external load on the IL thickness and the reduced length of the plastic zone is obtained. At the same time, the separation of the external load into elastic and plastic components is preserved. From the analysis of the experimental data, an estimate of the elastic limit of the energy product for the AV138 adhesive was obtained.

Elastic-Plastic Problem for a Stringer Plate with a Circular Hole

When calculating the strength of machines, structures and buildings with technological holes, it is important to take into account the plastic zones that emerge around the holes. However, the unknown shape and size of the plastic zone complicate the solution of elastic-plastic problems. This paper gives an approximate method and solution of the plane elastic-plastic problem of the distribution of stresses in a thin plate, reinforced with a regular system of stiffeners (stringers). The stringer plate under consideration has a circular hole, which is completely surrounded by the zone of plastic deformation. At infinity, the plate is subjected to a uniform tension along the stiffeners. A constant normal load is applied to the contour of the hole. The plate and stringer materials are assumed to be isotropic. The loading conditions are assumed to be quasi-static. It is assumed that the plate is in the plane-stressed state. Taken as the plasticity condition in the plastic zone is the Tresca-Saint-Venant plasticity condition. Methods of perturbation theory, analytic function theory, and the least squares method are used. The solution to the stated elastic-plastic problem consists of two stages. At the first stage, the stress-strain state for the elastic zone is found, and then the unknown interface between the elastic and plastic zones is determined using the least squares method. A closed system of algebraic equations has been constructed in each approximation, the numerical solution of which makes it possible to study the stress-strain state of a stringer plate, with the hole entirely surrounded by the plastic zone, as well as to determine the magnitudes of the concentrated forces that replace the action of the stringers. The interface between the elastic and plastic deformations has been found. The presented solution technique can be developed to solve other elastic-plastic problems. The solution obtained in this paper makes it possible to consider elastic-plastic problems for a stringer plate with other plasticity criteria.

Study and Computer-aided Design of Flux-cored Wire Rolling in Round Gauges

The mathematical model of the stress-strain state during flux-cored wire rolling in round gauges has been developed. Simulation was based on dividing the deformation zone into elementary volumes and simultaneous solution of the plasticity condition for porous materials and power static equilibrium equation inside the elementary volume. A distinctive feature of this model is taking into account the porous medium strain in the deformation zone. The experiments have confirmed the validity of the mathematical model for predicting the powder density and the energy-power characteristics of the process. Based on the developed mathematical model, the criteria and conditions for optimization were formulated, and the algorithm was developed for the automated design of technology for flux-cored wire rolling in round gauges. As an example of the obtained solutions implementation, the calculation of sintered copper flux-cored wire rolling technology was given.

Thick-walled spherical shell problem

Introduction. Cylindrical and spherical shells are extensively used in engineering. They face internal and/or external pressure and heat. Stresses and strains distribution in elastoplastic shells has been studied by many scientists. Numerous works involve the use of the von Mises yield conditions, maximum shear stress, maximum reduced stress. These condi- tions do not include the dependence on the first invariant of the stress tensor and the sign of the third invariant of the stress deviator. In some cases, it is possible to obtain numerical-analytical solutions for stresses, displacements and de- formations for bodies with spherical and cylindrical symmetry under axisymmetric thermal and force action.Materials and Methods. The problem on the state of a thick-walled elastoplastic shell is solved within the framework of the theory of small deformations. A plasticity condition is proposed, which takes into account the dependence of the stress tensor on three independent invariants, and also considers the sign of the third invariant of the stress deviator and translational hardening of the material. A disconnected thermoelastoplastic problem is being solved. To estimate the stresses in the region of the elastic state of a spherical shell, an equivalent stress is introduced, which is similar to the selected plasticity function. The construction of the stress vector hodograph is used as a method for verification of the stress state.Results. The problem has an analytical solution for linear plasticity functions. A solution is obtained when the strength- ening of the material is taken into account. Analytical and graphical relationships between the parameters of external action for the elastic or elastoplastic states of the sphere are determined. For a combined load, variants are possible when the plastic region is generated at the inner and outer boundaries of the sphere or between these boundaries.Discussion and Conclusions. The calculation results have shown that taking into account the plastic compressibility and the dependence of the plastic limit on temperature can have a significant impact on the stress and strain state of a hollow sphere. In this case, taking into account the first invariant of the stress tensor under the plasticity condition leads to the fact that not only the pressure drop between the outer and inner boundaries of the spherical shell, but the pressure values at these boundaries, can vary within a limited range. In this formulation of the problem, when there is only thermal action, the hollow sphere does not completely pass into the plastic state. The research results provide predicting the behavior of an object (a hollow sphere) that experiences centrally symmetric distributed power and thermal external influences.

MATHEMATICAL MODELING OF COLD PRESSING THE SHEET COMPOSITE

Statement of the problem. The article examines the problem of cold pressing, which is the most important technological component in the production of sheet composite, which is widely studied in the repair and construction works in the interior decoration of residential and industrial premises. The solution to this problem is carried out on the basis of a physical and mathematical model under the assumption that the rheological properties of the deformable medium correspond to the principles of ideal plasticity and a flat deformable state. Within the framework of the problem, in two dimensions of quasistatic compression between absolutely rigid parallel-approaching plates of a thin ideally plastic layer, the stress-strain state of a composite medium is studied. It is believed that in the absence of volumetric loads, the condition of incompressibility of the medium and the associated flow law is fulfilled. Based on the hypothesis of the linear distribution of tangential stresses over the thickness of the deformable layer, analytical expressions for the statistical and kinematic characteristics of the deformation are obtained, and the condition at the edges of the rough plates makes it possible to determine the coefficient of slip thorns, which makes it possible to control the pressing process.Results and conclusions. It was established that the components of the strain rate are directly proportional to the plate approach speed, and the normal stresses acting in the pressing direction are independent of the loading speed, decreasing in magnitude from the center to the periphery.Keywords: yield strength, pressing, plasticity condition, mathematical model.

Torsion of rods made of anisotropically hardening material under a linearized plasticity condition

В работе исследовано кручение стержней из анизотропно упрочняющегося жесткопластического материала. Получены интегралы, определяющие напряженное и деформированное состояния стержня при линеаризованном условии пластичности. Построены линии разрыва напряжений. The torsion of rods made of anisotropically hardening rigid-plastic material is studied. Integrals are obtained that determine the stress and strain States of the rod under the linearized plasticity condition. Stress discontinuity lines are constructed.

XFEM Fracture Analysis of 2-D Plastically Graded Domain With Thermo-Mechanical J-Integral

Many engineering components fail in the presence of service loads like thermal residual stresses and thermomechanical loading. An accurate evaluation of the fracture parameter (J-integral) at the crack tip is essential for the safe design of structures. In this work, a novel computational method called the Extended Finite Element Method (XFEM) has been implemented to analyze the plastically graded material (PGM) subjected to thermal and thermo-mechanical loading. For crack discontinuity modeling, a partition of unity enrichment concept can be employed with additional mathematical functions like Heaviside and branch enrichment for crack discontinuity and stress field gradient, respectively. The modeling of the stressstrain relationship of material has been done using the Ramberg-Osgood material model. The isotropic hardening and Von-Mises yield criteria have been considered to check the plasticity condition. The variation in plasticity properties for PGM has been modeled by exponential law. Further, the nonlinear discrete equation has been numerically solved using a Newton-Rhapson iterative scheme.

Torsion of inhomogeneous rods made of an ideal rigid plastic material under a linearized plasticity condition

В работе исследовано кручение неоднородных стержней из идеального жесткопластического материала. Получены интегралы, определяющие напряженное и деформированное состояния стержня при линеаризованном условии пластичности. Определено предельное состояние призматического стержня при кручении, найдены линии разрыва напряжений. The torsion of inhomogeneous rods made of an ideal rigid-plastic material is studied. Integrals are obtained that determine the stress and strain States of the rod under the linearized plasticity condition. The limit state of the prismatic rod during torsion is determined, and the stress break lines are found