Boundary Conditions in Problems of Plane Plastic Flow

• A. D. Tomlenov
Chapter

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

Shrinkage Dition Incompressibility

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References

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