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 three-dimensional state in the scope of the von Mises and Tresca yield condition. The chapter closes with classical failure and fracture hypotheses without the consideration of damage effects.
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Notes
- 1.
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 \(\sigma _\text {eff}:(\mathrm{{I}}{\negthinspace }\!\mathrm{{R}}^6 \rightarrow \mathrm{{I}}{\negthinspace }\!\mathrm{{R}}_+)\).
- 2.
In the general three-dimensional case \({\varvec{r}}\) hereby defines the direction of the vector \(\mathrm{{d}}\varvec{\varepsilon }^\text {pl}\), while the scalar factor defines the absolute value.
- 3.
Daniel Charles Drucker (1918–2001), US engineer.
- 4.
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 [8].
- 5.
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. 3.4b.
- 6.
Also signum function; from the Latin ‘signum’ for ‘sign’.
- 7.
The effective plastic strain is in the general three-dimensional case the function \(\varepsilon _\text {eff}^\text {pl}:(\mathrm{{I}}{\negthinspace }\!\mathrm{{R}}^6 \rightarrow \mathrm{{I}}{\negthinspace }\!\mathrm{{R}}_+)\). 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 \(\sqrt{\tfrac{2}{3}}\approx 0.816\).
- 8.
This is the volume-specific definition, meaning \(\left[ w^\text {pl}\right] =\tfrac{\text {N}}{\text {m}^2}\tfrac{\text {m}}{\text {m}} =\tfrac{\text {kg}\,\text {m}}{\text {s}^2\text {m}^2}\tfrac{\text {m}}{\text {m}}=\tfrac{\text {kg}\,\text {m}^2}{\text {s}^2\text {m}^3}=\tfrac{\text {J}}{\text {m}^3}\).
- 9.
Johann Bauschinger (1834–1893), German mathematician and engineer.
- 10.
An alternative expression for the kinematic hardening parameter is back-stress .
- 11.
William Prager (1903–1980), German engineer and applied mathematician.
- 12.
Hans Ziegler (1910–1985), Swiss scientist.
- 13.
Percy Williams Bridgman (1882–1961), American physicist.
- 14.
This formulation would be expected in a finite element code.
- 15.
It is useful for some applications (e.g. the calculation of derivative with respect to the stresses) to not consider the symmetry of the shear stress components and to work with nine stress components. These invariants are denoted by \(\underline{I}_i\) and \(\underline{J}_i\).
- 16.
The trace of a tensor is the sum of the diagonal elements.
- 17.
It should be noted that in the case of anisotropic materials, a hydrostatic stress state may result in a shape change, [8].
- 18.
Also called the hydrostatic stress; in the context of soil mechanics, the pressure \(p=-\sigma _{\text {m}}\) is also used.
- 19.
Albert Einstein (1879–1955), German theoretical physicist.
- 20.
See Sect. 3.1.3.3.
- 21.
Richard Edler von Mises (1883–1953), Austrian scientist and mathematician.
- 22.
Henri Édouard Tresca, (1814–1885), French mechanical engineer.
- 23.
Alternatively named as the maximum principal stress hypothesis.
- 24.
William John Macquorn Rankine (1820–1872), Scottish mechanical engineer, civil engineer, physicist and mathematician.
- 25.
Gabriel Léon Jean Baptiste Lamé (1795–1870), French mathematician.
- 26.
Claude-Louis Navier, (1785–1836), French engineer and physicist.
- 27.
Alternatively named as the maximum principal strain hypothesis.
- 28.
Adhémar Jean Claude Barré de Saint-Venant (1797–1886), French mechanician and mathematician.
- 29.
Carl Julius von Bach (1847–1931), German mechanical engineer.
- 30.
Alternatively named as the maximum strain energy hypothesis.
- 31.
Eugenio Beltrami (1835–1899), Italian mathematician.
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Öchsner, A. (2016). Elasto-Plastic Material Behavior. In: Continuum Damage and Fracture Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-287-865-6_3
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DOI: https://doi.org/10.1007/978-981-287-865-6_3
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