The objective of this post is the establish the basic formulations of a homogenized plasticity model. By ’homogenized’ we refer to the macroscopic description of polycrystalline aggregates in which the dislocation activity on individual slip systems is averaged over a large number of randomly oriented grains. In general, this theory is not applicable to single crystals.
In the von-Misses theory, the yield criterion is derived from the critical shear strain energy that initiates plastic flow. This leads to a yield surface that is a function of the second invariant of the deviatoric stress tensor:
\[\begin{split} f(\boldsymbol{\sigma},\boldsymbol{q}) &= \sqrt{2J_2} - \sqrt{\frac{2}{3}}\left(\sigma_y + H\alpha\right)\\ &= \sqrt{\boldsymbol{s}:\boldsymbol{s}} - \sqrt{\frac{2}{3}}\left(\sigma_y + H\alpha\right)\\ &= ||\boldsymbol{s}|| - \sqrt{\frac{2}{3}}\left(\sigma_y + H\alpha\right)\\ \end{split}\tag{1}\]
where the second invariant of the deviatoric stress tensor is defined as:
\[J_2 = \frac{1}{2}\boldsymbol{s}:\boldsymbol{s},\]
and the deviatoric stress as \(\boldsymbol{s} = \boldsymbol{\sigma} - \boldsymbol{1}\frac{1}{3}\textrm{tr} \boldsymbol{\sigma}\).
Consider a simple 1D problem in which the material is strained and undergoes a plastic flow. The following are the set of equations to describe this problem:
The constitutive equation:
\[\dot{\sigma} = E\left(\dot{\varepsilon} - \dot{\lambda}\frac{\partial f}{\partial \sigma}\right).\tag{2}\]
The yield surface:
\[\begin{split} f &= \sigma - \left(\sigma_y + H \alpha\right) = 0 \\ \dot{f} &= \dot{\sigma} - H\dot{\alpha} =0 \\ \end{split}.\tag{3}\]
The internal variable \(\alpha\) representing the equivalent plastic strain:
\[\alpha = \bar{\varepsilon}^p = \int_0^t \dot{\varepsilon}^p d \tau = \int_0^t \dot{\lambda} d\tau.\]
Assuming the plastic flow only, and putting all the equations together, we get:
\[E\frac{d\varepsilon}{d\lambda}\frac{d\lambda}{dt} - E\frac{d\lambda}{dt} - H\frac{d\lambda}{dt}.\tag{4}\]
which gives the scalar plastic multiplier: \[\lambda = \frac{E}{E+H}\varepsilon,\tag{5}\]
during the plastic flow.
Finally, the stress-strain curve becomes:
\[\sigma = \left\{ \begin{array}{l c l} E\varepsilon & \textrm{if} & \varepsilon < \frac{\sigma_y}{E} \\ E\left[\varepsilon - \frac{E}{E+H}\left(\varepsilon-\frac{\sigma_y}{E}\right)\right] & \textrm{if} & \varepsilon \geq \frac{\sigma_y}{E} \end{array} \right..\tag{6}\]
The two interesting cases emerge (in the plastic regime) when \(H=0\), then \(\sigma = \sigma_y\) and when \(H\rightarrow\infty\), then \(\sigma = E\varepsilon\).
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Let’s assume a simple power-law function and define the plastic strain rate as:
\[\dot{\varepsilon}^p = \dot{\gamma}_0 \left(\frac{\sigma}{s_0}\right)^{\frac{1}{m}} = \dot{\gamma}_0 \left(\frac{\sigma}{s_0}\right)^{n}\]
Substituting into a simple constitutive model, one arrives to the following:
\[\dot{\sigma} = E\left(\dot{\varepsilon}-\dot{\gamma_0}\left(\frac{\sigma}{s_0}\right)^n\right)\tag{7}\]
This represents a non-linear equation that can be discretized in time and solved using the Newton-Raphson method.
\[\frac{\sigma_{i+1}-\sigma_i}{\Delta t} = E \left(\frac{\Delta \varepsilon}{\Delta t} - \dot{\gamma}_0 \left(\frac{\sigma_{i+1}}{s_0}\right)^n\right)\tag{8}\]
\[\frac{E\dot{\gamma_0}}{s_0^n} \sigma_{i+1}^n + \frac{1}{\Delta t}\sigma_{i+1} - \left(E\frac{\Delta \varepsilon}{\Delta t} + \frac{\sigma_i}{\Delta t}\right) = 0\tag{9}\]
Using the Newton-Raphson method, we get
\[\begin{split} \sigma_{i+1}^{k+1} &= \sigma_{i+1}^k - \frac{f(\sigma^k_{i+1})}{f'(\sigma^k_{i+1})}\\ \sigma_{i+1}^{k+1} &= \sigma_{i+1}^k - \frac{\frac{E\dot{\gamma_0}}{s_0^n} \left(\sigma_{i+1}^k\right)^n + \frac{1}{\Delta t}\sigma_{i+1}^k - \left(E\frac{\Delta \varepsilon}{\Delta t} + \frac{\sigma_i}{\Delta t}\right)}{\frac{nE\dot{\gamma}_0}{s_0^n}\left(\sigma_{i+1}^k\right)^{n-1} + \frac{1}{\Delta t}} \end{split}\tag{10}\]