laws:vmvp
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laws:vmvp [2024/11/15 14:13] carlos [VMVP: damage formulations] |
laws:vmvp [2024/11/29 15:53] (current) carlos [VMVP: formulation generalities] |
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| $\Phi = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})-\sigma_{y}-R \leq 0$\\ | $\Phi = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})-\sigma_{y}-R \leq 0$\\ | ||
| - | where:\\ | + | >where: |
| - | * $\sigma_{y}$ is the yield stress of the material.\\ | + | >* $\sigma_{y}$ is the yield stress of the material. |
| - | * $R$ is the isotorpic hardening, calculated as the Voce iso. hardening law:\\ | + | >* $R$ is the isotorpic hardening, calculated as the Voce iso. hardening law: |
| - | $\dot{R} = -b\cdot (Q-R)\cdot \dot{p} \hspace{1cm} -> \hspace{1cm} R(p)=Q\cdot (1-\exp(-b\cdot p))$\\ where $p$ is the equivalent plastic strain $(-)$, and $\dot{p}$ is the equivalent plastic strain rate $(s^{-1})$\\ | + | >$\hspace{1cm} \dot{R} = -b\cdot (Q-R)\cdot \dot{p} \hspace{1cm} -> \hspace{1cm} R(p)=Q\cdot (1-\exp(-b\cdot p))$\\ where $p$ is the equivalent plastic strain $(-)$, and $\dot{p}$ is the equivalent plastic strain rate $(s^{-1})$\\ |
| Inheriting the classical Chaboche-type formulation, the backstress tensor $\underline{\mathbb{X}}$ is calculated as the sum of a total of $nAF$ user-defined Armstrong-Frederick (AF) backstress equations. \\ | Inheriting the classical Chaboche-type formulation, the backstress tensor $\underline{\mathbb{X}}$ is calculated as the sum of a total of $nAF$ user-defined Armstrong-Frederick (AF) backstress equations. \\ | ||
| Line 61: | Line 61: | ||
| > | > | ||
| > **option ivp=4: AFN** (Activation Function $\times$ Norton) | > **option ivp=4: AFN** (Activation Function $\times$ Norton) | ||
| - | >> This equation is new, and was formulated after a thorough analysis on the creep response of Incoloy 800H under lows-tress and high-temperature loadings. It is is intended to follow the classical Norton behavior, where the material reaches a steady-state creep rate $p_{\text{ss}}$. However, the phenomena such as solid-solution hardening or particle-strengthening may induce an initial creep hardening effect where the material can reach a creep rate lower that that of the steady state. To address this initial hardening, a phenomenological temperature and stress-dependent activation function has been added. The form of AFN is:\\ $\exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr] \times \langle \frac{\sigma_v}{K} \rangle ^{n}$ \\ Let us now walk you through the Activation Function particulars:\\ | + | >> This equation is new, and was formulated after a thorough analysis on the creep response of Incoloy 800H under lows-tress and high-temperature loadings. It is is intended to follow the classical Norton behavior, where the material reaches a steady-state creep rate $p_{\text{ss}}$. However, the phenomena such as solid-solution hardening or particle-strengthening may induce an initial creep hardening effect where the material can reach a creep rate lower that that of the steady state. To address this initial hardening, a phenomenological temperature and stress-dependent activation function has been added. The form of AFN is:\\ $\dot{p}= \exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr] \times \langle \frac{\sigma_v}{K} \rangle ^{n}$ \\ Let us now walk you through the Activation Function particulars:\\ |
| >>> __Activation Function__. The Activation Function is of the form: | >>> __Activation Function__. The Activation Function is of the form: | ||
| >>> $\exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr]$ | >>> $\exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr]$ | ||
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| >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k})^{-1}$ | >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k})^{-1}$ | ||
| >> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k},... \Bigr)$\\ | >> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k},... \Bigr)$\\ | ||
| - | > **dam_type = 3** Fully-coupled damage. | + | > **dam_type = 2** Fully-coupled damage. |
| >In this final case, the damage equations are included within the big Jacobian unknown set of equations, and are solved together with all the system in an implicit manner. It is a time-consuming and computationally heavy approach, where: | >In this final case, the damage equations are included within the big Jacobian unknown set of equations, and are solved together with all the system in an implicit manner. It is a time-consuming and computationally heavy approach, where: | ||
| >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k+1})^{-1}$ | >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k+1})^{-1}$ | ||
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| ^Line 01; format (2I5, 60A1)^^ | ^Line 01; format (2I5, 60A1)^^ | ||
| |IL|Law number| | |IL|Law number| | ||
| - | |ITYPE| 271| | + | |ITYPE| 282| |
| |COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing| | |COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing| | ||
laws/vmvp.1731676424.txt.gz · Last modified: 2024/11/15 14:13 by carlos
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