In contrast with the tensile test in which two of the principal stresses are zero, in a typical sheet process most elements will deform under membrane stresses σ1 and σ2, which are both non-zero. The third stress, σ3, perpendicular to the surface of the sheet is usually quite small as the contact pressure between the sheet and the tooling is generally very much lower than the yield stress of the material. As indicated above, we will make the simplifying assumption that it is zero and assume plane stress deformation, unless otherwise stated. If we also assume that the same conditions of proportional, monotonic deformation apply as for the tensile test, then we can develop a simple theory of plastic deformation of sheet that is reasonably accurate. We can illustrate these processes for an element as shown in Figure (a) for the uniaxial tension and Figure (b) for a general plane stress sheet process.
Stress and strain ratios
It is convenient to describe the deformation of an element, as in Figure (b), in terms of either the strain ratio β or the stress ratio α. For a proportional process, which is the only kind we are considering, both will be constant. The usual convention is to define the principal directions so that σ1 > σ2 and the third direction is perpendicular to the surface where σ3 = 0. The deformation mode is thus:
ε1 ; ε2 = βε1 ; ε3 = −(1 + β)ε1
σ1 ; σ2 = ασ1 ; σ3 = 0
The constant volume condition is used to obtain the third principal strain. Integrating thestrain increments in dε1 + dε2 + dε3 = 0 shows that this condition can be expressed in terms of the true or natural strains:
i.e. the sum of the natural strains is zero.
For uniaxial tension, the strain and stress ratios are β = −1/2 and α = 0.
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