Calculation of exact forces to cause plastic deformation in metal forming processes is often difficult. Exact solutions must be both statically and kinematically admissible. This means they must be geometrically self-consistent as well as satisfying stress equilibrium everywhere in the deforming body. Slip-line field analysis for plane strain deformation satisfies both and are therefore exact solutions. This topic is treated in Chapter 15. Upper and lower bounds are based on well-established principles [1, 2].

Frequently, it is difficult to make exact solutions and it is simpler to use limit theorems, which allows one to make analyses that result in calculated forces that are known to be either correct or too high or too low than the exact solution.

UPPERBOUNDS

The upper bound theorem states that any estimate of the forces to deform a body made by equating the rate of internal energy dissipation to the external forces will equal or be greater than the correct force. The analysis involves:

1. Assuming an internal flow field that will produce the shape change.

2. Calculating the rate at which energy is consumed by this flow field.

3. Calculating the external force by equating the rate of external work with the rate of internal energy consumption.