Book contents
- Frontmatter
- Contents
- Preface to Third Edition
- 1 Stress and Strain
- 2 Plasticity
- 3 Strain Hardening
- 4 Instability
- 5 Temperature and Strain-Rate Dependence
- 6 Work Balance
- 7 Slab Analysis and Friction
- 8 Upper-Bound Analysis
- 9 Slip-Line Field Analysis
- 10 Deformation-Zone Geometry
- 11 Formability
- 12 Bending
- 13 Plastic Anisotropy
- 14 Cupping, Redrawing, and Ironing
- 15 Forming Limit Diagrams
- 16 Stamping
- 17 Other Sheet-Forming Operations
- 18 Formability Tests
- 19 Sheet Metal Properties
- Index
- References
8 - Upper-Bound Analysis
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to Third Edition
- 1 Stress and Strain
- 2 Plasticity
- 3 Strain Hardening
- 4 Instability
- 5 Temperature and Strain-Rate Dependence
- 6 Work Balance
- 7 Slab Analysis and Friction
- 8 Upper-Bound Analysis
- 9 Slip-Line Field Analysis
- 10 Deformation-Zone Geometry
- 11 Formability
- 12 Bending
- 13 Plastic Anisotropy
- 14 Cupping, Redrawing, and Ironing
- 15 Forming Limit Diagrams
- 16 Stamping
- 17 Other Sheet-Forming Operations
- 18 Formability Tests
- 19 Sheet Metal Properties
- Index
- References
Summary
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 the 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 higher or lower 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 method of analysis is to
Assume an internal flow field that will produce the shape change;
Calculate the rate at which energy is consumed by this flow field; and
Calculate the external force by equating the rate of external work with the rate of internal energy consumption.
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- Chapter
- Information
- Metal FormingMechanics and Metallurgy, pp. 110 - 127Publisher: Cambridge University PressPrint publication year: 2007