Exercises 4f handbook jackhumechanicsofsheetmetalformingsecond

2 Sheet deformation processes

2.1 Introduction

In Chapter 1, the appropriate definitions for stress and strain in tensile deformation were introduced. The purpose now is to indicate how the true stress–strain curve derived from a tensile test can be applied to other deformation processes that may occur in typical sheet forming operations. A common feature of many sheet forming processes is that the stress perpendicular to the surface of the sheet is small, compared with the stresses in the plane of the sheet the membrane stresses. If we assume that this normal stress is zero, a major simplification is possible. Such a process is called plane stress deformation and the theory of yielding for this process is described in this chapter. There are cases in which the through-thickness or normal stress cannot be neglected and the theory of yielding in a three-dimensional stress state is described in an appendix. The tensile test is of course a plane stress process, uniaxial tension, and this is now reviewed as an example of plane stress deformation.

2.2 Uniaxial tension

We consider an element in a tensile test-piece in uniaxial deformation and follow the process from an initial small change in shape. Up to the maximum load, the deformation is uniform and the element chosen can be large and, in Figure 2.1, we consider the whole gauge section. During deformation, the faces of the element will remain perpendicular to each other as it is, by inspection, a principal element, i.e. there is no shear strain associated with the principal directions, 1, 2 and 3, along the axis, across the width and through the thickness, respectively. dt t w P d w 3 d l 1 2 l Figure 2.1 The gauge element in a tensile test-piece showing the principal directions. 14

2.2.1 Principal strain increments

During any small part of the process, the principal strain increment along the tensile axis is given by Equation 1.10 and is dε 1 = dl l 2.1 i.e. the increase in length per unit current length. Similarly, across the strip and in the through-thickness direction the strain increments are dε 2 = dw w and dε 3 = dt t 2.2

2.2.2 Constant volume incompressibility condition

It has been mentioned that plastic deformation occurs at constant volume so that these strain increments are related in the following manner. With no change in volume, the differential of the volume of the gauge region will be zero, i.e. dlwt = dl o w t o = 0 and we obtain dl × wt + dw × lt + dt × lw = 0 or, dividing by lwt, dl l + dw w + dt t = 0 i.e. dε 1 + dε 2 + dε 3 = 0 2.3 Thus for constant volume deformation, the sum of the principal strain increments is zero.

2.2.3 Stress and strain ratios isotropic material

If we now restrict the analysis to isotropic materials, where identical properties will be measured in all directions, we may assume from symmetry that the strains in the width and thickness directions will be equal in magnitude and hence, from Equation 2.3, dε 2 = dε 3 = − 1 2 dε 1 In the previous chapter we considered the case in which the material was anisotropic where dε 2 = Rdε 3 and the R-value was not unity. We can develop a general theory for anisotropic deformation, but this is not necessary at this stage. Sheet deformation processes 15