Yielding in plane stress

The stresses required to yield a material element under plane stress will depend on the current hardness or strength of the sheet and the stress ratio α. The usual way to define the strength of the sheet is in terms of the current flow stress σf. The flow stress is the stress at which the material would yield in simple tension, i.e. if α = 0. This is illustrated in the true stress–strain curve in Figure. Clearly σf depends on the amount of deformation to which the element has been subjected and will change during the process. For the moment, we shall consider only one instant during deformation and, knowing the current value of σf the objective is to determine, for a given value of α, the values of σ1 and σ2 at which the element will yield, or at which plastic flow will continue for a small increment. We consider here only the instantaneous conditions in which the strain increment is so small that the flow stress can be considered constant. In Chapter 3 we extend this theory for continuous deformation.
There are a number of theories available for predicting the stresses under which a material element will deform plastically. Each theory is based on a different hypothesis about material behaviour, but in this work we shall only consider two common models and apply them to the plane stress process described by Equations ε1; ε2 = βε1; ε3 = −(1 + β)ε1. Over the years, many researchers have conducted experiments to determine how materials yield. While no single theory agrees exactly with experiment, for isotropic materials either of the models presented here are sufficiently accurate for approximate models.

With hindsight, common yielding theories can be anticipated from knowledge of the nature of plastic deformations in metals. These materials are polycrystalline and plastic flow occurs by slip on crystal lattice planes when the shear stress reaches a critical level. To a first approximation, this slip which is associated with dislocations in the lattice is insensitive to the normal stress on the slip planes. It may be anticipated then that yielding will be associated with the shear stresses on the element and is not likely to be influenced by the average stress or pressure. It is appropriate to define these terms more precisely.