Slip-line field theory

Definition

Consider a plane strain deformation in the $(x, y)$-plane of a rigid–perfectly plastic isotropic material with shear yield stress $k > 0$. In the plastic region the in-plane Cauchy stress components satisfy the yield condition below. $$\frac{(\sigma_{xx} - \sigma_{yy})^2}{4} + \sigma_{xy}^2 = k^2$$ It is the plane strain form of the von Mises and Tresca criteria. Then there are a function $p$, the hydrostatic pressure, and an angle $\phi$ with $$\sigma_{xx} = -p - k \sin 2\phi, \quad \sigma_{yy} = -p + k \sin 2\phi, \quad \sigma_{xy} = k \cos 2\phi.$$ The $\alpha$-lines are the curves whose tangent makes the angle $\phi$ with the $x$-axis, and the $\beta$-lines are their orthogonal trajectories. These two orthogonal families of curves are the directions of maximum shear stress, and together they form the slip-line field. Slip-line field theory is the solution of plane strain plasticity problems by constructing this field. It uses that the slip lines are the characteristics of the hyperbolic system formed by equilibrium and the yield condition.