Abstract
The continuum mechanics basics for the one-dimensional bar will be compiled at the beginning of this chapter. The yield condition, the flow rule, the hardening law and the elasto-plastic modulus will be introduced for uniaxial, monotonic loading conditions. Within the scope of the hardening law, the description is limited to isotropic hardening, which occurs for example for the uniaxial tensile test with monotonic loading. For the integration of the elasto-plastic constitutive equation, the incremental predictor-corrector method is generally introduced and derived for the fully implicit and semi-implicit backward-Euler algorithm. On crucial points the difference between one- and three-dimensional descriptions will be pointed out, to guarantee a simple transfer of the derived methods to general problems. Calculated examples and supplementary problems with short solutions serve as an introduction for the theoretical description.
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Notes
- 1.
The case of unloading or alternatively load reversal will not be regarded at this point due to simplification reasons.
- 2.
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 \(\sigma _\text{eff}:(\text{I}\!\text{R}^6 \rightarrow \text{I}\!\text{R}_+)\).
- 3.
In the general three-dimensional case r hereby defines the direction of the vector \(\text{d}\varvec{\varepsilon }^\text{pl}\), while the scalar factor defines the absolute value.
- 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 [3].
- 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. 11.2b.
- 6.
Also signum function; from the Latin ‘signum’ for ‘sign’.
- 7.
The effective plastic strain is in the general three-dimensional case the function \(\varepsilon _\text{eff}^\text{pl}:(\text{I}\!\text{R}^6 \rightarrow \text{I}\!\text{R}_+)\). In the here regarded 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 \(\varepsilon _\text{eff}^\text{pl}=\sqrt{\frac{2}{3}\sum \Delta \varepsilon _{ij}^\text{pl}\sum \Delta \varepsilon _{ij}^\text{pl}}\), which considers the lateral contraction at uniaxial stress problems in the plastic area via the factor \(\frac{2}{3}\). 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 \(\sqrt{\frac{2}{3}}\approx 0.816\).
- 8.
This is the volume-specific definition, meaning \(\left[w^\text{pl}\right]=\frac{\text{N}}{\text{m}^2}\frac{\text{m}}{\text{m}} =\frac{\text{kg}\,\text{m}}{\text{s}^2\text{m}^2}\frac{\text{m}}{\text{m}}=\frac{\text{kg}\,\text{m}^2}{\text{s}^2\text{m}^3}=\frac{\text{J}}{\text{m}^3}\).
- 9.
In the general three-dimensional case one talks about the elasto-plastic stiffness matrix \(\varvec{C}^\text{elpl}\).
- 10.
In the general case with six stress and strain components (under consideration of the symmetry of the stress and strain tensor) an obvious relation only exists between effective stress and effective plastic strain. In the one-dimensional case however these parameters reduce to: \(\sigma _\text{eff}=|\sigma |\) and \(\varepsilon _\text{eff}^\text{pl}=|\varepsilon ^\text{pl}|\).
- 11.
The explicit Euler procedure or polygon method (also Euler-Cauchy method) is the most simple procedure for the numerical solution of an initial value problem. The new stress state results according to this procedure in \(\sigma _{n+1}=\sigma _n+E^\text{elpl}_n\Delta \varepsilon \), whereupon the initial value problem can be named as \(\frac{\text{d} \sigma }{\text{d}\varepsilon }=E^\text{elpl}(\sigma ,\varepsilon )\) with \(\sigma (\varepsilon _0)=\sigma _0\).
- 12.
In the general three-dimensional case the relation is applied on the stress vector and the increment of the strain vector: \(\varvec{\sigma }_{n+1}^\text{trial}=\varvec{\sigma }_n+\varvec{C}\Delta \varvec{\varepsilon }_n\).
- 13.
At this point within the notation it is formally switched from \(\text{d}\lambda \) to \(\Delta \lambda \). Therefore the transition from the differential to the incremental notation occurs.
- 14.
from the Latin ‘residuus’ for left or remaining.
- 15.
The Newton method is usually used as follows for a one-dimensional function: \(x^{(i+1)}=x^{(i)}-\left(\frac{\text{d}f}{\text{d}x}(x^{(i)})\right)^{-1}\times f(x^{(i)})\).
- 16.
Also referred to as consistent elasto-plastic tangent modulus matrix, consistent tangent stiffness matrix or algorithmic stiffness matrix.
- 17.
At this point it is switched from \(\text{d}\lambda \) to \(\Delta \lambda \).
- 18.
At this point for the considered linear bar elements a constant strain distribution per element results. In general, the strain results as a function of the element coordinates which is usually evaluated on the integration points. Therefore, one would in the general case normally define a strain \(vector \varvec{\varepsilon }\) per element, which combines the different strain values on the integration points. This however is unnecessary for a linear bar element. A scalar strain or alternative stress value is enough for the description.
- 19.
Hereby the energy per unit volume is considered.
- 20.
- 21.
This plane has to stand vertically on the \(\sigma{-}|\varepsilon ^\text{pl}|\) plane. For a tensile test the plane has to go through the limit curve in the area \(\sigma > 0\). For a compression test the according straight line from the area \(\sigma < 0\) has to be chosen.
- 22.
In the considered example with linear hardening, \(\tilde{E}\) is constant in the elastic range (increment 1–3) and in the plastic range (increment 4–10) and therefore not a function of \(u_2\). In the general case however \(\tilde{E}\) has to be differentiated as well.
- 23.
One considers that in both sections or alternatively elements, the stress and strain are identical.
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Öchsner, A., Merkel, M. (2013). Plasticity. In: One-Dimensional Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31797-2_11
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DOI: https://doi.org/10.1007/978-3-642-31797-2_11
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