scholarly journals Solution Behavior in the Vicinity of Characteristic Envelopes for the Double Slip and Rotation Model

2020 ◽  
Vol 10 (9) ◽  
pp. 3220 ◽  
Author(s):  
Yao Wang ◽  
Sergei Alexandrov ◽  
Elena Lyamina
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Strain Rate Tensor ◽  
Axially Symmetric ◽  
Natural Boundary ◽  
Rigid Plastic ◽  
Solution Behavior ◽  
Normal Distance ◽  
Approach Infinity ◽  
Rotation Model

The boundary conditions significantly affect solution behavior near rough interfaces. This paper presents general asymptotic analysis of solutions for the rigid plastic double slip and rotation model in the vicinity of an envelope of characteristics under plane strain and axially symmetric conditions. This model is used in the mechanics of granular materials. The analysis has important implications for solving boundary value problems because the envelope of characteristics is a natural boundary of the analytic solution. Moreover, an envelope of characteristics often coincides with frictional interfaces. In this case, the regime of sticking is not possible independently of the friction law chosen. It is shown that the solution is singular in the vicinity of envelopes. In particular, the profile of the velocity component tangential to the envelope is described by the sum of the constant and square root functions of the normal distance to the envelope in its vicinity. As a result, some components of the strain rate tensor approach infinity. This finding might help to develop an efficient numerical method for solving boundary value problems and provide the basis for the interpretation of some experimental results.

scholarly journals An axially symmetric forced convection problem

1975 ◽  
Vol 20 (1) ◽  
pp. 1-17
Author(s):  
J. A. Belward
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Convection Heat Transfer ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Original Equation ◽  
Transfer Model ◽  
Axially Symmetric ◽  
Simple Diffusion ◽  
Diffusion Convection

AbstractA simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

scholarly journals Solution Behavior Near Very Rough Walls under Axial Symmetry: An Exact Solution for Anisotropic Rigid/Plastic Material

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 184
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Pierre-Yves Manach
Keyword(s):  
Boundary Value Problem ◽  
Axial Symmetry ◽  
Friction Surface ◽  
Plastic Material ◽  
Boundary Value ◽  
System Of Equations ◽  
Rigid Plastic ◽  
Solution Behavior ◽  
Rigid Plastic Material ◽  
Friction Surfaces

Rigid plastic material models are suitable for modeling metal forming processes at large strains where elastic effects are negligible. A distinguished feature of many models of this class is that the velocity field is describable by non-differentiable functions in the vicinity of certain friction surfaces. Such solution behavior causes difficulty with numerical solutions. On the other hand, it is useful for describing some material behavior near the friction surfaces. The exact asymptotic representation of singular solution behavior near the friction surface depends on constitutive equations and certain conditions at the friction surface. The present paper focuses on a particular boundary value problem for anisotropic material obeying Hill’s quadratic yield criterion under axial symmetry. This boundary value problem represents the deformation mode that appears in the vicinity of frictional interfaces in a class of problems. In this respect, the applied aspect of the boundary value problem is not essential, but the exact mathematical analysis can occur without relaxing the original system of equations and boundary conditions. We show that some strain rate and spin components follow an inverse square rule near the friction surface. An essential difference from the available analysis under plane strain conditions is that the system of equations is not hyperbolic.

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