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.

Analysis of the Trajectory of a Hydraulic Fracture near the Seam Working and the Conditions for its Straight-line Growth

Model of the geomechanical state of a disc hydraulic fracture propagating in the solid rocks near the seam working is based on the methods of solid mechanics and fracture mechanics. Stress field in the coal-rock mass containing in-seam working and growing hydraulic fracture was constructed as a result of solving an elastoplastic problem, in which the area of plasticity is the extremely stressed zones of the edge parts of the seam. The stress field in the edge parts was determined in the course of the numerical solution of three boundary value problems of the seam limiting state. The criteria for the onset of the limiting state are the general criterion of the Coulomb — Mohr limiting state for the formation and a special criterion for the limiting state for its contact with the rock mass. By replacing the extremely stressed zones with the stresses acting on their contact with the rock mass, the elastoplastic problem is reduced to the second external boundary value problem of the theory of elasticity, which is solved by the method of boundary integral equations. At the relatively low fluid pressures in the pumping unit, the trajectory of the hydraulic fracture is a smooth curved line of the small length with a significant deviation of its ends from the direction of the seed crack. With increasing fluid pressure, the crack length increases, and the deviation from the direction of the seed crack decreases. There are fluid pressures at which the crack propagates in a straight line and practically does not change its original direction. The straight-line trajectory of the crack in the vicinity of the working located at different depths corresponds to a point on the graph of the dependence of the relative length of the crack on the fluid relative pressure. This graph is a straight line.

Geometrically exact elastoplastic rods: determination of yield surface in terms of stress resultants

This work addresses the determination of yield surfaces for geometrically exact elastoplastic rods. Use is made of a formulation where the rod is subjected to an uniform strain field along its arc length, thereby reducing the elastoplastic problem of the full rod to just its cross-section. By integrating the plastic work and the stresses over the rod’s cross-section, one then obtains discrete points of the yield surface in terms of stress resultants. Eventually, Lamé curves in their most general form are fitted to the discrete points by an appropriate optimisation method. The resulting continuous yield surfaces are examined for their scalability with respect to cross-section dimensions and also compared with existing analytical forms of yield surfaces.

INVESTIGATION OF LOCAL UNLOADING OF AN ELEMENT IN A FINITE ELEMENT CONTINUUM

The subcritical elastoplastic deformation and the fracturing of an element of a finite element continuum in the Ansys Workbench complex are considered. When solving the elastoplastic problem of the subcritical deformation, a finite element with the failure criterion reached is selected. In a pre-fracture state of the element, the nodal forces provided by the interaction with an adjacent element are determined using the Ansys Workbench internal procedure. The following step is the consideration of the varying stress-strain state of the body during the element destruction. The elastoplastic problem is solved in the conditions of simple unloading of the body surface adjacent to the destructible element while maintaining the external load corresponding to the destruction initiation. When implementing the local unloading, a possibility of the new plastic region formation and the partial unloading are studied. As a result, the stress-strain state of the body at the beginning of local unloading is not the same as that at the end of the process. The proposed approach differs from the “element killing” procedure when the element stiffness after the failure criterion reached is assumed to be close to zero. The paper provides solutions to the problems of deformation of elastic and elastoplastic plates with a side cut taking into account their element destruction.

http://www.mining-science.ru/download/2020/3/04.pdf

Introduction. Reliable forecasts of pillars geomechanical state are required to ensure rhythmic and safe work when mining a coal bed. Research aim is to construct a state model of the coal pillar located between the headways, based on the fundamental methods of elasticity theory and mechanics of a granular media, carry out a computational experiment within the model, and analyse the results. Methodology. The stress field in the coal pillar has been constructed in the course of solving the elastoplastic problem. By replacing the ultimately stressed marginal zone of the bed with the stresses which act within the zone, the problem has been reduced to the second exterior boundary value problem of elasticity theory and has been solved by the boundary element method. Ordinary and special Coulomb–Mohr criteria simultaneously fulfilled for the coal bed and rock mass contact are the criterion of the limit state onset. Actual pillar load is determined by integrating the vertical stress curve along the bed roof, which has been obtained from elastoplastic problem solution, while the ultimate load is determined from the condition that the whole pillar is in ultimately stressed state. Results. The dependence between the safety factor of the pillar between two identical headways, determined by V. D. Shevyakov method, and the growth of its width represents a graph in the form of a monotonically increasing curve. The curve flattens as soon as the depth increases. Summary. The results from the developed model of coal rock mass geomechanical state can be successfully used as coal pillar strength forecasts.

Anisotropic Antiplane Elastoplastic Problem

In this work we solve an anisotropic antiplane elastoplastic problem about stress state in a body weakened by a hole bounded by a piecewise-smooth contour. We give the conservation laws which allowed us to reduce calculations of stress components to a contour integral over the contour of the hole. The conservation laws allowed us to find the boundary between the elastic and plastic areas

Elastoplastic Equilibrium of a Hollow Cylinder from an Inhomogeneous Perfectly Plastic Material

The article deals with the axisymmetric elastoplastic problem for a hollow thick-walled cylinder (plane deformed state), loaded from the inside and outside by uniform pressures proportional to one parameter. The material is considered to be perfectly plastic, with the elastic modulus and yield strength generally are arbitrary functions of the radius. In addition, the material is considered to be incompressible in both plastic and elastic zones. On the basis of the criteria for the plasticity of Huber - Mises and Tresca - Saint-Venant, the radius at which the first plastic deformations occur is determined. It is shown that, depending on the functions of the inhomogeneity of elastic and plastic parameters and loads, the occurrence of plastic deformations is possible both on the surfaces and on the inner walls of the cylinder.

Elastoplastic Equilibrium of a Hollow Thick-Walled Radially Inhomogeneous Ball

The article deals with the axisymmetric elastoplastic problem for a hollow thick-walled ball (plane deformed state), loaded from the inside and outside by uniform pressures proportional to one parameter. The material is considered to be perfectly plastic, with the elastic modulus and yield strength generally are arbitrary functions of the radius. In addition, the material is considered to be incompressible in both plastic and elastic zones. On the basis of the criteria for the plasticity of Huber - Mises and Tresca - Saint-Venant, the radius at which the first plastic deformations occur is determined. It is shown that, depending on the functions of the inhomogeneity of elastic and plastic parameters and loads, the occurrence of plastic deformations is possible both on the surfaces and on the inner walls of the ball.

Existence for dislocation-free finite plasticity

This note addresses finite plasticity under the constraint that plastic deformations are compatible. In this case, the total elastoplastic deformation of the medium is decomposed as y = ye ○ yp, where the plastic deformation yp is defined on the fixed reference configuration and the elastic deformation ye is a mapping from the varying intermediate configuration yp(Ω). Correspondingly, the energy of the medium features both Lagrangian (plastic, loads) and not Lagrangian contributions (elastic). We present a variational formulation of the static elastoplastic problem in this setting and show that a solution is attained in a suitable class of admissible deformations. Possible extensions of the result, especially in the direction of quasistatic evolutions, are also discussed.