Constant volume incompressibility condition

We may summarize the tensile test process for an isotropic material in terms of the strain increments and stresses in the following manner: dε 1 = dl l ; dε 2 = − 1 2 dε 1 ; dε 3 = − 1 2 dε 1 2.4a and σ 1 = P A ; σ 2 = 0; σ 3 = 0 2.4b

2.2.4 True, natural or logarithmic strains

It may be noted that in the tensile test the following conditions apply: • the principal strain increments all increase smoothly in a constant direction, i.e. dε 1 always increases positively and does not reverse; this is termed a monotonic process; • during the uniform deformation phase of the tensile test, from the onset of yield to the maximum load and the start of diffuse necking, the ratio of the principal strains remains constant, i.e. the process is proportional; and • the principal directions are fixed in the material, i.e. the direction 1 is always along the axis of the test-piece and a material element does not rotate with respect to the principal directions. If, and only if, these conditions apply, we may safely use the integrated or large strains defined in Chapter 1. For uniaxial deformation of an isotropic material, these strains are ε 1 = ln l l ; ε 2 = ln w w = − 1 2 ε 1 ; ε 3 = ln t t o = − 1 2 ε 1 2.5

2.3 General sheet processes plane stress

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 σ 1 and σ 2 , which are both non-zero. The third stress, σ 3 , 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 2.2a for the uniaxial tension and Figure 2.2b for a general plane stress sheet process.

2.3.1 Stress and strain ratios

It is convenient to describe the deformation of an element, as in Figure 2.2b, 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 16 Mechanics of Sheet Metal Forming