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v0.13.2 |
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v0.13.2 |
In the following document, we define functions, \(\psi(\varepsilon^e,\alpha)\), \(f(\sigma,\mathbf{q})\) and \(\Psi(\sigma,\mathbf{q})\) which are free Helmholtz energy, yield function and flow potential, respectively. Note that free Helmholtz energy is defined by stains and kinematic internal state variables, whereas yield function and flow potential is defined by fluxes (stress tensor and fluxes work conjugated to internal kinematic state variables). In this implantation, we assume that all functions are convex. Also, is assumed that energy potential and flow potential is positive and takes zero value at zero strain and zero internal variable states. It can be shown that for such defined problem, the first law of thermodynamics and non-negative dissipation of energy is automatically satisfied, see [19] [44]. Moreover, in the following implementation is assumed that all functions are smooth enough that appropriate derivatives could be calculated.
In the following implementation, the backward-Euler difference scheme is applied to solve set nonlinear system of equations with unilateral constrains. Admissible stress state is enforced by Lagrange multiplier method and (Khun-Tucker) loading/unloading conditions, see [44]. That lead to well known Closet point projection algorithm.
Following [44] can be shown that plastic loading unloading conditions can be exclusively characterised in terms of trial state
\[ \varepsilon^{e,\textrm{trial}}_{n+1} := \varepsilon_{n+1} - \varepsilon^p_n \]
\[ \alpha^\textrm{trial}_{n+1} := \alpha_n \]
\[ \sigma^\textrm{trial}_{n+1} := \frac{\partial \psi}{\partial \varepsilon^e}(\varepsilon^{e,\textrm{trial}}_{n+1},\alpha_n) \]
\[ \mathbf{q}^\textrm{trial}_{n+1} := \frac{\partial \psi}{\partial \alpha}(\varepsilon^{e,\textrm{trial}}_{n+1},\alpha_n) \]
\[ f^\textrm{trial}_{n+1} := f(\sigma^\textrm{trial}_{n+1},\mathbf{q}_n) \]
then loading/unloading conditions are given
\[ \begin{split} f^\textrm{trial}_{n+1} < 0,\quad\textrm{elastic step}\,\to\,\Delta\gamma_{n+1} = 0 \\ f^\textrm{trial}_{n+1} \geq 0,\quad\textrm{plastic step}\,\to\,\Delta\gamma_{n+1} > 0 \end{split} \]
If first condition is true \(f^\textrm{trial}_{n+1} < 0\), trial stress is solution at integration point. If second condition is true, plastic strains and values of internal state variables are calculated using Closet point projection algorithm.
\[ \varepsilon^e_{n+1} := \varepsilon_{n+1} - \varepsilon^p_{n+1} \]
\[ \sigma_{n+1} := \frac{\partial \psi}{\partial \varepsilon^e}(\varepsilon^e_{n+1},\alpha_{n+1}) \]
\[ \mathbf{q}_{n+1} := \frac{\partial \psi}{\partial \alpha}(\varepsilon^e_{n+1},\alpha_{n+1}) \]
\[ f_{n+1}:=f(\sigma_{n+1},\mathbf{q}_{n+1}) \]
Flow vector
\[ \mathbf{n}_{n+1}:=\frac{\partial \Psi}{\partial \sigma}(\sigma_{n+1},\mathbf{q}_{n+1}) \]
Hardening law:
\[ \mathbf{h}_{n+1}:=-\frac{\partial \Psi}{\partial \mathbf{q}}(\sigma_{n+1},\mathbf{q}_{n+1}) \]
\[ \mathbf{R}_{n+1}:= \left\{ \begin{array}{c} -\varepsilon^p_{n+1}+\varepsilon^p_n \\ -\alpha_{n+1}+\alpha_n \\ f_{n+1} \end{array} \right\} + \Delta\gamma_{n+1} \left\{ \begin{array}{c} \mathbf{n}_{n+1} \\ \mathbf{h}_{n+1} \\ 0 \end{array} \right\} \]
\[ \left[ \begin{array}{ccc} -1+\gamma \left( \frac{\partial\mathbf{n}}{\partial\sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \varepsilon^p} + \frac{\partial\mathbf{n}}{\partial\mathbf{q}}\frac{\partial^2 \psi}{\partial\alpha\varepsilon^p} \right) & \gamma \left( \frac{\partial\mathbf{n}}{\partial\mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha^2} + \frac{\partial\mathbf{n}}{\partial\sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \alpha} \right) & \mathbf{n} \\ \gamma \left( \frac{\partial\mathbf{h}}{\partial\sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \varepsilon^p} + \frac{\partial\mathbf{h}}{\partial\mathbf{q}}\frac{\partial^2 \psi}{\partial\alpha\partial\varepsilon^p} \right) & -1+\gamma \left( \frac{\partial\mathbf{h}}{\partial\mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha^2} + \frac{\partial\mathbf{h}}{\partial\sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \alpha} \right) & \mathbf{h}\\ \frac{\partial f}{\partial \sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \varepsilon^p} + \frac{\partial f}{\partial \mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha\partial\varepsilon^p} & \frac{\partial f}{\partial \mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha^2} + \frac{\partial f}{\partial \sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial\alpha} & 0 \end{array} \right]_n \left\{ \begin{array}{c} \delta \varepsilon^p \\ \delta \alpha \\ \delta \gamma \end{array} \right\}_{n+1} = \mathbf{R}_n \]
where \(\varepsilon^p_{n+1}=\varepsilon^p_n + \delta\varepsilon^p_{n+1}\), \(\alpha^p_{n+1}=\alpha_n + \delta\alpha_{n+1}\) and \(\Delta\gamma_{n+1} = \Delta\gamma_n+\delta\gamma_{n+1}\). Usually it is assumed free energy split, i.e.
\[ \psi(\varepsilon^e,\alpha) = \psi^{\varepsilon}(\varepsilon^e) + \psi^\alpha(\alpha) \]
in that case linearized residual takes the form
\[ \left[ \begin{array}{ccc} -1+\Delta\gamma \frac{\partial\mathbf{n}}{\partial\sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \varepsilon^p} & \Delta\gamma \frac{\partial\mathbf{n}}{\partial\mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha^2} & \mathbf{n} \\ \Delta\gamma \frac{\partial\mathbf{h}}{\partial\sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \varepsilon^p} & -1+\Delta\gamma \frac{\partial\mathbf{h}}{\partial\mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha^2} & \mathbf{h}\\ \frac{\partial f}{\partial \sigma}\frac{\partial^2 \psi}{\partial \varepsilon^e \partial \varepsilon^p} & \frac{\partial f}{\partial \mathbf{q}}\frac{\partial^2 \psi}{\partial \alpha^2} & 0 \end{array} \right]_n \left\{ \begin{array}{c} \delta \varepsilon^p \\ \delta \alpha \\ \delta \gamma \end{array} \right\}_{n+1} = -\mathbf{R}_n \]
\[ \chi_{n+1} := \left\{ \begin{array}{c} \delta \varepsilon^p \\ \delta \alpha \\ \delta \gamma \end{array} \right\}_{n+1} = - \mathbf{A}^{-1} \mathbf{R}_n \]
\[ \delta \sigma := \mathbf{C}^e(\delta \varepsilon - \delta \varepsilon^p) \]
where \(\mathbf{C}^e = \frac{\partial^2 \psi^e}{\partial {\varepsilon^e}^2}\) and \(\delta \varepsilon = \textrm{sym}[ \nabla \delta \mathbf{u}]\)
\[ \left\{ \begin{array}{c} \mathbf{E}^p \\ \mathbf{L}^p \\ \mathbf{Y}^p \end{array} \right\} \delta \varepsilon = - \mathbf{A}^{-1} \left\{ \begin{array}{c} \Delta\gamma \frac{\partial \mathbf{n}}{\partial \sigma}\mathbf{C}^e \\ \Delta\gamma \frac{\partial \mathbf{h}}{\partial \sigma}\mathbf{C}^e \\ \frac{\partial f}{\partial \sigma}\mathbf{C}^e \end{array} \right\} \delta \varepsilon \]
\[ \delta \varepsilon^p = \mathbf{E}^p \delta \varepsilon \]
\[ \delta \alpha = \mathbf{L}^p \delta \varepsilon \]
\[ \delta \gamma = \mathbf{Y}^p \delta \varepsilon \]
\[ \delta \sigma = \mathbf{C}^e(\mathbf{1} - \mathbf{E}^p)\delta \varepsilon \]
\[ \delta \sigma = \mathbf{C}^{ep}\delta \varepsilon \]
where \(\mathbf{C}^{ep} = \mathbf{C}^e-\mathbf{C}^e\mathbf{E}^p\)
For more details and easy access to all source files related to this see group Small Strain Plasticity.
