Verifying the Movable Elastoplastic Boundary Method by Using Galin’s Problem

Galin’s solution for the problem of biaxial tension of a plate with a hole completely covered by the plastic region appears to be a pearl recognized by the world scientific community. This solution serves as a test for all sorts of approximate approaches to solving elastoplastic problems, including the semi-analytical iterative method being developed by the author, focused on solving more complex problems such as the Kirsch problem in the elastoplastic formulation. The proposed iterative approach for a semi-analytical solution involves an explicit analytical expression for stresses in the plastic region and an iterative numerical solution in the elastic region with a refined border. The paper shows the convergence of the results based on the iterative procedure for the elastoplastic region boundary approaching its analytical position, which follows from the analytical solution of Galin’s elastoplastic problem. Consideration has also been given to obtaining results on the determination of the boundary between the elastic and plastic regions using a competing approximate perturbation method. The advantage of the proposed method lays in not limited modifications in parameters due to the requirement for small differences while formulating a problem from the axisymmetric case as seen in the perturbation method.

Study on Formation Mechanism of Zonal Disintegration of Surrounding Rock in Deep Tunnel

With the construction of underground rock engineering, the surrounding rock in deep tunnels appears zonal disintegration of fracture and intact zones alternate distribution, which is a special engineering geological phenomenon. This study establishes a partitioned fracture model under the coupling of high in-situ stress and osmotic pressure, and identify the key influencing factors of the fracture model. Furthermore, a stress intensity factor (SIF) of initial cracks on surrounding rock elastoplastic boundary is derived using the transformation of complex functions. Considering the high seepage pressure of the surrounding rocks, a zonal fracture initiation criterion is established combined with the analysis of redistributed stress fields. Finally, the obtained criterion is embedded into an extended finite element method (XFEM) platform for numerical simulation. Taking the maximum circumferential tensile stress as a cracking criterion, the propagation trajectory of rock cracks is traced by contour methods. Calculation results have realized the modelling of a whole process of crack initiation, propagation, and formation. And the established criterion has be verified.

Determination of the stress state of a hardening disk under the influence of temperature

Как известно при нагревании твердые тела, в частности металлы, испытывают температурные деформации, связанные с эффектом линейного расширения. При этом, несмотря на то, что эти деформации малы, соответствующие им напряжения могут оказаться достаточно большими, зачастую превосходящими предел текучести материала. В связи с этим для описания напряженно-деформированных состояний тел, находящихся под действием высоких температур, необходимо учитывать неупругое поведение материалов. Определению напряжений и деформаций в упругопластических задачах посвящено множество работ, в том числе исследования [1-10]. В некоторых из них [1], [5-10] рассматривается температурное воздействие на тела различной конфигурации. В настоящей работе решается задача об определении осесимметричного поля напряжений в плоском диске при воздействии на него точечного источника тепла (например, точечная сварка). Материал диска моделируется упрочняющейся упругопластической средой. Задача решалась в рамках плоско-напряженного состояния методом малых возмущений. В аналитическом виде получены соотношения, описывающие распределение полей напряжений в упругой и пластической областях деформирования. В качестве условий сопряжения решений на упругопластической границе использовались условия неразрывности радиальной и окружной компонент тензора напряжений и радиальной компоненты вектора перемещений. As you know, when heated, solids, in particular metals, experience thermal deformations associated with the effect of linear expansion. Moreover, in spite of the fact that these strains are small, the corresponding stresses can be quite large, often exceeding the yield stress of the material. In this regard, to describe the stress-strain states of bodies exposed to high temperatures, it is necessary to take into account the inelastic behavior of materials. Determination of stresses and strains in elastoplastic problems has been the subject of many works, including studies [1-10]. Some of them [1], [5-10] consider the temperature effect on bodies of various configurations. In this work, we solve the problem of determining the axisymmetric stress field in a flat disk when exposed to a point heat source (for example, spot welding). The disc material is modeled by a hardening elastoplastic medium. The problem was solved within the plane-stressed state by the method of small perturbations. In an analytical form, relations are obtained that describe the distribution of stress fields in elastic and plastic deformation regions. The conditions of continuity of the radial and circumferential components of the stress tensor and the radial component of the displacement vector were used as conditions for conjugation of solutions on the elastoplastic boundary.

Elastoplastic Bending of the Console with Transverse Force

In the article an elastoplastic boundary for the console being bent with transverse force when the point of force is not situated in the centroid of transverse section was built with the use of the conservation laws. In this case bending moments and torques appear of the console. The case when the point of force is situated in the centroid of transverse section is considered in the previous works of the authors. In the work an infinite system of conservation laws has been built that allows us to reduce the problem of calculating elastoplastic boundary to a few quadratures, at the outer contour of transverse section. At that the contour can be random piecewise smooth. It is assumed that the lateral surface of the console is free from strains and is in its plastic condition

Solution of tasks for elastoplastic transversal isotropic bodies

In this article: the theorem has been proven for existence and uniqueness of the generalized solution of elastoplastic boundary problem based on the theory of plastic yielding of transversal isotropic bodies from the loading surface in the deformation and stress space.