laws:vmvp
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laws:vmvp [2024/11/15 14:07] carlos |
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. \\ | ||
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| > | > | ||
| > **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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| >> Similar to the previous law, Kachanov formulation is introduced as: | >> Similar to the previous law, Kachanov formulation is introduced as: | ||
| >> $\dot{D}_{c}=k_3 \Bigl[\frac{\mathcal{Y(k_{4}\cdot \underline{\sigma})}}{S_{c}} \Bigr]^{S_{ce}} \frac{1}{(1-D)^{k_k}} $ | >> $\dot{D}_{c}=k_3 \Bigl[\frac{\mathcal{Y(k_{4}\cdot \underline{\sigma})}}{S_{c}} \Bigr]^{S_{ce}} \frac{1}{(1-D)^{k_k}} $ | ||
| - | >> where $S_{c}$, $S_{ce}$, $k_3$, $k_4$, $k_k$ are material constants. In particular, $S_c$ is conceived as $f(T)$. | + | >> where $S_{c}$, $S_{ce}$, $k_3$, $k_4$, $k_k$ are material constants. In particular, $S_c$ is conceived as $f(T)$.\\ |
| + | > **idam=2**: IfW Creep-Fatigue damage formulation | ||
| + | >This creep-fatigue damage law was developed by IfW institute in Aachen University. It came to our knowledge after being used by N. K. Karthik in the context of his PhD. project on the life prediction of metallic components subjected to creep-fatigue loadings. It is inspired in the Graham-Walles formulation, and is of the form: | ||
| + | > $\dot{D}=K_{\text{D}} J_{2}\Bigl( \underline{\tilde{\sigma}} - \underline{\mathbb{X}} \Bigr) + |\dot{T}|K_{\text{Td}}p^{m_{\text{Td}}}$ | ||
| + | > where $K_{\text{Td}}$, $K_{\text{Td}}$ and $m_{\text{Td}}$ are material parameters. | ||
| ===== Input file ===== | ===== Input file ===== | ||
| ==== Parameters defining the type of constitutive law ==== | ==== Parameters defining the type of constitutive law ==== | ||
| Line 120: | Line 124: | ||
| ^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.1731676030.txt.gz · Last modified: 2024/11/15 14:07 by carlos
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