Worked example stress state

Referring to Figure 2.3, the plastic work done on a unit volume of material deformed in the tensile test to a true strain of ε 1 where σ 2 = σ 3 = 0 will be, from Equation 2.15b, W vol. = ε 1 dW vol. = ε 1 σ 1 dε 1 2.16 i.e. the work done per unit volume is equal to the area under the true stress–strain curve, shown shaded in Figure 2.3.

2.7 Work hardening hypothesis

In Section 2.5 it was shown that at a particular instant in a plane stress process where the flow stress, σ f , was known, the stresses and the ratio of the strain increments for a small deformation could be determined. To model a process we need to be able to follow the deformation along the given loading path as the flow stress changes. Clearly we would need to know the strain hardening characteristic of the material as determined, for example, by the true stress–strain curve in the tensile test. It has been found by experiment that the flow stress increases in any process according to the amount of plastic work done during this process; i.e. in two different processes, if the work done in each is the same, the flow stress at the end of each process will be the same regardless of the stress path. This statement is only true for monotonic processes that follow the conditions given in Section 2.2.4; if there is a reversal in the process, the flow stress cannot be predicted by any simple theory. In a plane stress, proportional process, we can plot the relation between each of the non-zero stresses and its strain as shown in Figure 2.11. From Equation 2.15b, the sum of the shaded areas shown is the total work done per unit volume of material in the process. According to this work-hardening hypothesis, the flow stress at the end of this process is that given by the tensile test curve, Figure 2.3, when an equal amount of work has been done, i.e. when the sum of the areas in Figure 2.11 is equal to the area under the curve in Figure 2.3. s e 2 e e 1 s 2 s 1 Figure 2.11 Stress–strain curves for the principal directions 1 and 2 for an element deforming in a plane stress procedure in which σ 2 = ασ 1 . The way in which this work-hardening hypothesis is implemented in any analysis is described in the next section. Sheet deformation processes 25