True, natural or logarithmic strains

a b Tensile test Plane stress s 3 = 0, e 3 = −e 1 2 s 1 , e 1 s 2 = 0, e 2 = −e 1 2 s 3 = 0, e 3 = − 1 + be 1 s 1 , e 1 s 2 = as 1 e 2 = be 1 Figure 2.2 Principal stresses and strains for elements deforming in a uniaxial tension and b a general plane stress sheet process. principal directions so that σ 1 σ 2 and the third direction is perpendicular to the surface where σ 3 = 0. The deformation mode is thus: ε 1 ; ε 2 = βε 1 ; ε 3 = −1 + βε 1 2.6 σ 1 ; σ 2 = ασ 1 ; σ 3 = 0 The constant volume condition is used to obtain the third principal strain. Integrating the strain increments in Equation 2.3 shows that this condition can be expressed in terms of the true or natural strains: ε 1 + ε 2 + ε 3 = 0 2.7 i.e. the sum of the natural strains is zero. For uniaxial tension, the strain and stress ratios are β = −12 and α = 0.

2.4 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 σ f . 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 2.3. Clearly σ f 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 σ f the objective is to determine, for a given value of α, the values of σ 1 and σ 2 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 2.6. 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. Sheet deformation processes 17