Calculation of exact forces to cause plastic deformation in metal forming processes is often difficult. Exact solutions must be both statically and kinematically admissible. That means they must be geometrically self-consistent as well as satisfying required stress equilibrium everywhere in the deforming body. Frequently it is simpler to use limit theorems that allow 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.

Lower bounds are based on satisfying stress equilibrium, while ignoring geometric self-consistency. They give forces that are known to be either too low or correct. As such they can assure that a structure is “safe.” Conditions in which η = 0 are lower bounds. Upper-bound analyses on the other hand predict stress or forces that are known to be too large. These are usually more important in metal forming. Upper bounds are based on satisfying yield criteria and geometric self-consistency. No attention is paid to satisfying equilibrium.

UPPER BOUNDS

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.