Uniform sheet deformation processes

An instant in the plane stress deformation of a work-hardening material was considered. We now apply the theory to some region of a sheet undergoing uniform, proportional deformation as shown in Figure . If the undeformed sheet, of initial thickness t0 is marked with a grid of circles of diameter d0 or a square mesh of pitch d0 as shown in Figure (a), then during uniform deformation, the circles will deform to ellipses of major and minor axes d1 and d2 respectively. If the square grid is aligned with the principal directions, it will become rectangular as shown in Figure (b). The thickness is denoted by t . At the instant shown in Figure (b), the deformation stresses are σ1 and σ2.

Tension as a measure of force in sheet forming
In sheet processes, deformation occurs as the result of forces transmitted through the sheet. The force per unit width of sheet is the product of stress and thickness and in Figure (c) is represented by, T = σt where, T , is known as the tension, traction or stress resultant. Because this is the product of the current thickness t as well as the current stress σ, it is the appropriate measure of force and will be used throughout this work in modelling processes. The term, tension, will be used even though this suffers from the disadvantage that the force is not always
a tensile force. If the tension is negative, it indicates a compressive force. This is not a serious problem as in plane stress sheet forming, almost without exception, one tension will be positive, i.e. the sheet is always pulled in one direction. It is impractical to forms sheet by pushing on the edge; the expression used by practical sheet formers is that ‘you cannot push on the end of a rope’. In the convention used here, the principal direction 1 is that in which the principal stress has the greatest (most positive) value, and the major tension T1 = σ1t will always be positive. In stretching processes, the minor tension T2 = σ2t is tensile or positive. In other processes, the minor tension could be compressive and in some cases the thickness will increase. If T2 is compressive and large in magnitude, wrinkling may be a problem. In discussing true stress in Section , it was shown that for most real materials, strain-hardening continues, although at a diminishing rate, and true stress does not reach a maximum. As tension includes thickness, which in many processes will diminish, T may reach a maximum; this limits the sheet’s ability to transmit load and is one of the reasons for considering tension in any analysis.

Strain distributions
In the study of any process, we usually determine first the strain over the part. This can be done by measuring a grid as in Figure 3.1, or by analysis of the geometric constraint exerted on the part. An example is the deep drawing process in Figure (a) and in the Introduction, Figure I.9. As the process is symmetric about the axis, we need only consider the strain at points on a line as shown in Figure (b). Plotting these strains in the principal strain space, Figure (c), gives the locus of strains for a particular stage in the process. As the process continues, this locus will expand, but not necessarily uniformly; some points may stop straining, while others go on to reach a process limit. For any process, there will be a characteristic strain pattern, as shown in Figure (c). This is sometimes known as the ‘strain signature’. Considerable information can be
obtained from such a diagram and the way it is analysed is outlined in the following section.

Strain diagram
The individual points on the strain locus in second Figure (c) can be obtained from measurements of a grid circles as shown in first Figure . (If a square grid is used, the analysis method is outlined in Appendix A.2.) If the major and minor axes are measured and the current thickness determined, the analysis is as follows.