Abstract
This chapter introduces first the basic equations of plasticity theory. The yield condition, the flow rule, and the hardening rule are introduced. After deriving and presenting these equations for the one-dimensional stress and strain state, the equations are generalized for a two-component \(\sigma \)-\(\tau \) stress state. Based on these basic equations, the concept of effective stress and strain as well as the elasto-plastic modulus is introduced. The chapter finishes with an introduction to two different damage concepts, i.e. the Lemaitre and Gurson damage model, which are derived for the one-dimensional case.
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- 1.
Siméon Denis Poisson (1781–1840), French mathematician, geometer, and physicist.
- 2.
More modern devises are contactless laser or video based extensometers.
- 3.
Thomas Young (1773–1829), English polymath.
- 4.
Robert Hooke (1635–1703), English natural philosopher, architect and polymath.
- 5.
The case of unloading or alternatively load reversal will be treated separately in Sect. 2.7.
- 6.
If the unit of the yield criterion equals the stress, \(f(\sigma )\) represents the equivalent stress or effective stress. In the general three-dimensional case the following is valid under consideration of the symmetry of the stress tensor \(\sigma _\text {eff}:(\text {I}\!\text {R}^6 \rightarrow \text {I}\!\text {R}_+)\).
- 7.
A two-dimensional stress state would comprise in its general form, for example, the three components \(\sigma _x\), \(\sigma _y\) and \(\tau _{xy}\).
- 8.
Henri Édouard Tresca (1814–1885), French mechanical engineer.
- 9.
Richard Edler von Mises (1883–1953), Austrian scientist and mathematician.
- 10.
In the general three-dimensional case \({{\varvec{r}}}\) hereby defines the direction of the vector \(\text {d}\varvec{\varepsilon }^\text {pl}\), while the scalar factor defines the absolute value.
- 11.
Daniel Charles Drucker (1918–2001), US engineer.
- 12.
A formal alternative derivation of the associated flow rule can occur via the Lagrange multiplier method as extreme value with side-conditions from the principle of maximum plastic work [12].
- 13.
In the general three-dimensional case the image vector of the plastic strain increment has to be positioned upright and outside oriented to the yield surface, see Fig. 2.3b.
- 14.
Also signum function; from the Latin ‘signum’ for ‘sign’.
- 15.
The effective plastic strain is in the general three-dimensional case the function \(\varepsilon _\text {eff}^\text {pl}:(\text {I}\!\text {R}^6 \rightarrow \text {I}\!\text {R}_+)\). In the one-dimensional case, the following is valid: \(\varepsilon _\text {eff}^\text {pl}=\sqrt{\varepsilon ^\text {pl}\varepsilon ^\text {pl}}=|\varepsilon ^\text {pl}|\). Attention: Finite element programs optionally use the more general definition for the illustration in the post processor, this means \(\varepsilon _\text {eff}^\text {pl}=\sqrt{\tfrac{2}{3}\sum \Delta \varepsilon _{ij}^\text {pl}\sum \Delta \varepsilon _{ij}^\text {pl}}\), which considers the lateral contraction at uniaxial stress problems in the plastic area via the factor \(\tfrac{2}{3}\). However in pure one-dimensional problems without lateral contraction, this formula leads to an illustration of the effective plastic strain, which is reduced by the factor \(\sqrt{\tfrac{2}{3}}\approx 0.816\).
- 16.
This is the volume-specific definition, meaning \(\left[ w^\text {pl}\right] =\tfrac{\text {N}}{\text {m}^2}\tfrac{\text {m}}{\text {m}} =\tfrac{\text {kg}\,\text {m}}{\text {s}^2\text {m}^2}\tfrac{\text {m}}{\text {m}}=\tfrac{\text {kg}\,\text {m}^2}{\text {s}^2\text {m}^3}=\tfrac{\text {J}}{\text {m}^3}\).
- 17.
Johann Bauschinger (1834–1893), German mathematician and engineer.
- 18.
An alternative expression for the kinematic hardening parameter is back-stress.
- 19.
William Prager (1903–1980), German engineer and applied mathematician.
- 20.
Hans Ziegler (1910–1985), Swiss scientist.
- 21.
Sometimes called the equivalent stress.
- 22.
The yield surfaces for \(D>0\) look like ellipses but because of the \(\cosh \) function, they are not from a mathematical point of view classified as ellipses.
- 23.
The first component of the stress deviator is given by \(s_x=\tfrac{2}{3}\sigma _x-\tfrac{1}{3}\left( \sigma _y+\sigma _z\right) \).
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© 2014 Springer-Verlag Berlin Heidelberg
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Öchsner, A. (2014). Continuum Mechanics of Plasticity. In: Elasto-Plasticity of Frame Structure Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44225-8_2
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DOI: https://doi.org/10.1007/978-3-662-44225-8_2
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