Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

The Limits of Applicability of the Method of Discontinuous Solutions in the Study of Pipe Drawing Processes

Introduction. Non-contact deformation of the workpiece material, which occurs along the boundaries of the deformation zone, is one of the main factors determining the energy-power parameters of pipe reduction processes. The most widespread practice in the design of metal forming processes is the method of discontinuous solutions, which makes it quite simple to take into account non-contact deformation in numerical simulation of processes. However, for most processes in the technical literature there are no systematic practical recommendations on the application of this method, which inevitably leads to a mismatch of theoretical principles and practice. The aim of the work is to determine the limits of applicability of the method of discontinuous solutions for processes of faultless drawing of pipes through a conical die, depending on the geometric parameters of the workpiece, tool, as well as the degree of deformation and hardening of the processed material. Research Methods. The model of the deformation zone for the process of flawless drawing is considered in two versions: by the method of discontinuous solutions and taking into account non-contact bends of the pipe wall. From the condition of the balance of the shear forces acting on the conditional shear surface and the bending moments caused by the bending of the pipe wall, under various deformation conditions, the boundary values of the thickness parameter are determined, at which it is advisable to carry out numerical simulation of the drawing processes using the discontinuous solution method. In this case, the calculations are performed separately for two sections of the deformation zone corresponding to the bending of the pipe wall at the entrance to and exit from the die. Results and discussions. The numerical implementation of the obtained dependences showed that at the entrance to the deformation zone, the boundary value of the thickness parameter increases with an increase in the taper angle of the die and the hood for the transition, but decreases with an increase in the anti-tension stress and the thickness parameter of the initial workpiece. At the exit from the deformation zone, the boundary value of the thick-walled parameter increases with an increase in the taper angle of the die and decreases with an increase in the stretch coefficient for the transition and the thick-walled parameter of the initial billet. If the parameter of the thickness of the initial billet exceeds the boundary value, then in numerical modeling it is advisable to use the method of discontinuous solutions. If it does not exceed, then other methods and models should be used. The results of a theoretical study can be used in the design of pipe drawing processes.