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Boundary Conditions in Problems of Plane Plastic Flow

  • A. D. Tomlenov

Abstract

Plane plastic flow is described by the equilibrium conditions
$$\frac{{\partial \sigma _x }}{{\partial x}} + \frac{{\partial \tau _{yx} }}{{\partial y}} = 0,$$
(1)
$$\frac{{\partial \sigma _y }}{{\partial y}} + \frac{{\partial \tau _{xy} }}{{\partial x}} = 0;$$
(2)
by the plasticity condition
$$\left( {\sigma _x - \sigma _y } \right)^2 + 4\tau _{xy}^2 = 4k^2 ;$$
(3)
by the condition of coaxial alignment of the stress and deformation velocities
$$\frac{{\sigma _y - \sigma _x }}{{2\tau _{xy} }} = \frac{{\frac{{\partial v}}{{\partial y}} - \frac{{\partial u}}{{\partial x}}}}{{\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}}} = \tan 2\alpha ;$$
(4)
and by the incompressibility condition
$$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0,$$
(5)
where σxand σy are the normal stress components, τxy = τyx is the tangential stress component, k is the plastic constant, u and v are the velocity vector coordinates with respect to the x and y axes, α is the angle which the slip line of the first set of curves forms with the x axis, and x and y are the point coordinates.

Keywords

Plastic Flow Tangential Stress Slip Line Plastic Region Plasticity Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Consultants Bureau, New York 1971

Authors and Affiliations

  • A. D. Tomlenov

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