11: Fundamental Concepts in Structural Plasticity
- Page ID
- 21545
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Plastic properties of the material were already introduced briefly earlier in the present notes. The critical slenderness ratio of column is controlled by the yield stress of the material. The subsequent buckling of column in the plastic range requires the knowledge of the hardening curve. These two topics were described in Chapter 8. In Chapter 9 the concept of the ultimate strength of plates was introduced and it was shown that the yield stress is reached first along the supported or clamped edges and the plastic zones spread towards the plate center, leading to the loss of stiffness and strength. In the present lecture the above simple concepts will be extended and formalized to prepare around for the structural applications in terms of the limit analysis. There are five basic concepts in the theory of plasticity:
- Yield condition
- Hardening curve
- Incompressibility
- Flow rule
- Loading/unloading criterion
All of the above concept will first be explained in the 1-D case and then extended to the general 3-D case.
- 11.1: Hardening Curve and Yield Curve
- This page covers the engineering stress-strain curve derived from tensile tests on steel and aluminum specimens, highlighting concepts like the proportionality limit and the 0.02% yield point. It examines total strain components and presents empirical models, including the swift hardening law. Simplified stress-strain models, such as rigid-perfectly plastic materials for limit analysis, are also discussed, with an indication of future extensions to three-dimensional cases.
- 11.2: Loading/Unloading Condition
- This page explains the plastic flow rule for rigid-perfectly plastic materials under one-dimensional stress. It details how plastic strain rates relate to stress, identifying positive and negative flow conditions linked to yield stresses. It also clarifies that zero flow occurs within the elastic range between these stresses. Additionally, the page discusses the consistent unloading trajectory on a stress-strain graph, regardless of the unloading case.
- 11.3: Incompressibility
- This page covers plastic incompressibility in metals, explaining that their volume stays constant during plastic deformation. It describes the relationship between gauge length and cross-sectional area changes, leading to a strain definition. The discussion includes a three-dimensional analysis, establishing that the sum of the strain rate tensor's diagonal components must be zero, yielding a Poisson ratio of 0.5 for plastics, compared to 0.3 for metals.
- 11.4: Yield Condition
- This page covers the von Mises yield condition in three-dimensional stress analysis, detailing the yielding equation, its application in principal coordinate systems, and plane stress scenarios. It compares the von Mises condition with uniaxial conditions, emphasizing how stress influences yielding. Additionally, the page introduces equivalent stress and strain rate concepts vital for finite element analysis, explaining their formulations and integration methods for equivalent strain evaluation.
- 11.5: Isotropic and Kinematic Hardening
- This page examines material behavior under uniaxial stress, focusing on simplified stress and strain rates. It introduces isotropic hardening, connecting yield conditions to equivalent plastic strain and deriving hardening functions from uniaxial tension tests. The relationship between stress and strain is described using a power law, contrasting isotropic with kinematic hardening, the latter relating to yield surface shifts relevant for cyclic loading.
- 11.6: Flow Rule
- This page covers the flow rule for rigid perfectly plastic materials, detailing how the strain rate vector relates to stress through the equation \(\dot{\epsilon}_{ij} = \dot{\lambda} \frac{\partial F(\sigma_{ij})}{\partial \sigma_{ij}}\). It explains that the strain rate direction is normal to the yield surface, provides two components of strain rate under plane stress conditions with a fixed ratio, and discusses the implications of transverse plain strain on stress component relationships.
- 11.7: Derivation of the Yield Condition from First Principles (Advanced)
- This page covers the derivation and importance of stress-strain relationships for elastic materials, focusing on Hook's law and the components of strain energy. It explains that strain energy density is path-independent and presents key equations. It also introduces Huber's yield criterion, indicating yielding at a critical distortional energy density, and discusses the connection between stress components under uniaxial tension and the generalized von Mises criterion.
- 11.8: Tresca Yield Condition
- This page explores the connection between shear stress and uniaxial tension in materials, emphasizing how stress states are influenced by plane orientation. It presents Henri Tresca's maximum shear stress condition and yield criterion, supported by graphical representations. Additionally, the effects of hydrostatic pressure on yielding are examined, showing that under uniform pressure, neither the Tresca nor von Mises criteria indicate yielding.
- 11.9: Experimental Validation
- This page reviews the von Mises and Tresca yield criteria for material yielding under stress, outlining the methods used to derive stress states from tests on thin-walled tubes. It highlights the lack of a definitive preference for either criterion due to overlapping results and emphasizes their similarities, particularly regarding shear stresses and shape distortion.
- 11.10: Example of the Design against First Yield
- This page covers the safety and design of pressure vessels and piping systems, specifically focusing on thick pipes under internal pressure. It explores stresses in cylindrical coordinates, analyzing radial and circumferential stresses in elastic conditions.