Book contents
- Frontmatter
- Contents
- Preface
- 1 Stress and strain
- 2 Elastic and inelastic material behaviour
- 3 Yield
- 4 Plastic flow
- 5 Collapse load theorems
- 6 Slip line analysis
- 7 Work hardening and modern theories for soil behaviour
- Appendices
- A Non-Cartesian coordinate systems
- B Mohr circles
- C Principles of virtual work
- D Extremum principles
- E Drucker's stability postulate
- F The associated flow rule
- G A uniqueness theorem for elastic–plastic deformation
- H Theorems of limit analysis
- I Limit analysis and limiting equilibrium
- Index
- References
G - A uniqueness theorem for elastic–plastic deformation
Published online by Cambridge University Press: 23 November 2009
- Frontmatter
- Contents
- Preface
- 1 Stress and strain
- 2 Elastic and inelastic material behaviour
- 3 Yield
- 4 Plastic flow
- 5 Collapse load theorems
- 6 Slip line analysis
- 7 Work hardening and modern theories for soil behaviour
- Appendices
- A Non-Cartesian coordinate systems
- B Mohr circles
- C Principles of virtual work
- D Extremum principles
- E Drucker's stability postulate
- F The associated flow rule
- G A uniqueness theorem for elastic–plastic deformation
- H Theorems of limit analysis
- I Limit analysis and limiting equilibrium
- Index
- References
Summary
The concept of the uniqueness of a solution is an essential requirement to the well-posed nature of a boundary value problem. A uniqueness theorem assures us that there is only one solution possible for the governing set of equations subject to appropriate boundary conditions. In EG we have discussed a uniqueness theorem in the context of the linear theory of elasticity. With linear theories in mechanics and physics, the development of a proof of uniqueness of solutions to boundary value problems and initial boundary value problems is well established. Comprehensive discussions of these topics are given in many texts on mathematical physics and on the theory of partial differential equations and also discussed in recent volumes by Selvadurai (2000a,b). The question that arises in the context of plasticity focuses on the development of a uniqueness theorem for what is basically a non-linear problem. This is not a straightforward issue, even with regard to certain situations involving non-linear behaviour of linear elastic materials. Examples that illustrate the concept of non-uniqueness of elasticity solutions can be readily found in problems dealing with elastic buckling of structural elements such as beam-columns and shallow shells under lateral loads. In these categories of problem the structure can exhibit multiple equilibrium states corresponding to the same level of loading. The purpose of the discussion given below is then to address the basic question of what constraints should be imposed, specifically regarding plastic stress–strain relations, in order that the solution to a particular boundary value problem is unique.
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- Information
- Plasticity and Geomechanics , pp. 263 - 268Publisher: Cambridge University PressPrint publication year: 2002