laws:vmvp
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laws:vmvp [2024/11/15 10:58] carlos [Numerical model] |
laws:vmvp [2024/11/29 15:53] (current) carlos [VMVP: formulation generalities] |
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| 3D isotropic viscoplastic damage law that enables the modeling of non-classical creep responses via Graham-Walles and a modified activation function-Norton viscosity function. | 3D isotropic viscoplastic damage law that enables the modeling of non-classical creep responses via Graham-Walles and a modified activation function-Norton viscosity function. | ||
| - | ==== Numerical model ==== | + | === Overview === |
| This law was implemented in the context of C.Rojas-Ulloa's PhD. project (01/2021-12/2025) on the modeling of the non-classical long-term creep response of Incoloy 800H (see [[https://doi.org/10.1016/j.camwa.2023.12.002|(C.Rojas-Ulloa et al., 2024)]] for more details). This colaw is based in the work of Hélène Morch ([[laws:chab|CHAB]], a von-Mises yield function combined with a Norton-type viscosity function, Kachanov-Lemaitre creep-fatigue damage and high flexibility for introducing parameters as $f(T)$). **VMVP** incorporates two new viscoplastic functions (Graham-Walles & AFN), a simple //IfW// creep-fatiogue damage formulation, and new parameter interpolation methods available for material parameters. | This law was implemented in the context of C.Rojas-Ulloa's PhD. project (01/2021-12/2025) on the modeling of the non-classical long-term creep response of Incoloy 800H (see [[https://doi.org/10.1016/j.camwa.2023.12.002|(C.Rojas-Ulloa et al., 2024)]] for more details). This colaw is based in the work of Hélène Morch ([[laws:chab|CHAB]], a von-Mises yield function combined with a Norton-type viscosity function, Kachanov-Lemaitre creep-fatigue damage and high flexibility for introducing parameters as $f(T)$). **VMVP** incorporates two new viscoplastic functions (Graham-Walles & AFN), a simple //IfW// creep-fatiogue damage formulation, and new parameter interpolation methods available for material parameters. | ||
| - | === VMVP: formulation generalities=== | + | ==== VMVP: formulation generalities==== |
| The yield surface is defined by the von-Mises yield criterion: an isotropic $J_{2}$-type function of the form:\\ | The yield surface is defined by the von-Mises yield criterion: an isotropic $J_{2}$-type function of the form:\\ | ||
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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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| >> This term is important in presence of thermal gradients $\dot{T}$, and if we introduce the parameter $C_i$ as function of temperature.\\ | >> This term is important in presence of thermal gradients $\dot{T}$, and if we introduce the parameter $C_i$ as function of temperature.\\ | ||
| - | === VMVP: viscoplasticity === | + | ==== VMVP: viscoplasticity ==== |
| As a viscoplastiv law, **VMVP** includes a total of 3 viscoplastic functions. | As a viscoplastiv law, **VMVP** includes a total of 3 viscoplastic functions. | ||
| > **option ivp=1: Norton law** | > **option ivp=1: Norton law** | ||
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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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| >>>> where $p_{21}, p_{22}, p_{23}$ are user-defined material parameters. | >>>> where $p_{21}, p_{22}, p_{23}$ are user-defined material parameters. | ||
| - | === VMVP: damage formulations === | + | ==== VMVP: damage formulations ==== |
| Damage computation is added to enable the prediction of the effects of microcracks and its evolution towards the final fracture of the material. To use this formulation, the user must first define the damage methodology (dam_type) to be used in the simulation. A brief explanation of the avalable possibilities is given below:\\ | Damage computation is added to enable the prediction of the effects of microcracks and its evolution towards the final fracture of the material. To use this formulation, the user must first define the damage methodology (dam_type) to be used in the simulation. A brief explanation of the avalable possibilities is given below:\\ | ||
| > **dam_type = 0** No damage | > **dam_type = 0** No damage | ||
| >In this case, no damage is considered within the finite element simulations. Therefore | >In this case, no damage is considered within the finite element simulations. Therefore | ||
| > · $\tilde{\underline{\sigma}}_{k+1}=\underline{\sigma}_{k+1}$. | > · $\tilde{\underline{\sigma}}_{k+1}=\underline{\sigma}_{k+1}$. | ||
| - | > · $D=0$ | + | > · $D=0$\\ |
| > **dam_type = 1** uncoupled damage | > **dam_type = 1** uncoupled damage | ||
| - | > Here damage is calculated in an uncoupled manner (e.g., at the end of every %k+1% viscoplastic loop), but the stress is not affected by damage. | + | > Here damage is calculated explicitly (e.g., at the end of every %k+1% viscoplastic loop), but the stress is not affected by damage. |
| >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1}$ | >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1}$ | ||
| - | >> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, \D_{k},... \Bigr)$ | + | >> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k},... \Bigr)$\\ |
| + | > **dam_type = 2** semi-coupled damage | ||
| + | >In this case, damage is also computed explicitly. However, the effective stress is also computed, as a function of the damage in the previous configuration. i.e.: | ||
| + | >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k})^{-1}$ | ||
| + | >> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k},... \Bigr)$\\ | ||
| + | > **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: | ||
| + | >> · $\underline{\tilde{\sigma}}_{k+1}=\underline{\sigma}_{k+1} \cdot (1-D_{k+1})^{-1}$ | ||
| + | >> · $D_{k+1}=f \Bigl( \underline{\sigma}_{k+1}, D_{k+1},... \Bigr)$\\ | ||
| + | In the case that damage is considered within the simulation (i.e., **dam_type>0**), the user must subsequently define a damage intiation criteria (dam_init). A total of three options are given: | ||
| + | > **daminit=0** | ||
| + | > In this case, damage is set to be initiated together with plastic deformation. It is not advisable, as it often results in a huge damage overestimation.\\ | ||
| + | > **daminit=1**: plastic work level criterion | ||
| + | > Following H. Morch's developments, this criteiron is based in an adjusted plastic work level calculation where damage is initiated after a critical $w_c$ plastic work level is reached. The equation describing the accumulated plastic work considers terms for both isotorpic and kinematic hardening, and is of the form: | ||
| + | >> $w(p)=\displaystyle\int_{0}^{t} \Big[ \frac{A}{m} p^{\frac{1-m}{m}} R(p)\dot{p} \Big]dt + \displaystyle\sum_{i=1}^{nAF}\frac{3}{2}\underline{\mathbb{X}}_i : \underline{\mathbb{X}}_i$ | ||
| + | >>damage is initiated when $w(p)\geq w_c$ | ||
| + | >>where $w_c$, $A$ and $m$ are adjustable material parameters. Particularly, $w_c$ and $A$ can be introduced as $f(T)$.\\ | ||
| + | > **daminit=3** Plastic deformation level | ||
| + | > This last criterion consist simply in the activation of the damage evolution mechanism when a user-defined critical plastic deformation level $p_c$ is reached. It can be introduced as a constant or as a temperature-dependent Arrhenius function.\\ | ||
| + | Once the previous constrains are defined, the user mus now define the damage law. In **VMVP**, two damage laws are given for the user to choose from: | ||
| + | > **idam=1**: Lemaitre-Kachanov Creep-Fatigue damage formulation | ||
| + | >Implemented after H. Morch's work, this formulation has proven high reliability when facing complex creep-fatigue loadings. It consiste in the direct sum of the contributions of creep (Kachanov) and fatigue (Lemaitre) damage formulations. These are described hereafter: | ||
| + | >> __Fatigue damage: Lemaitre law__ | ||
| + | >> This fatigue law is independent of time, and depends more on the. It is of the form: | ||
| + | >> $\dot{D}_{f}=k_1 \Bigl[\frac{\mathcal{Y(k_{2}\cdot \underline{\sigma})}}{S_{f}} \Bigr]^{S_{fe}} \dot{p}$ | ||
| + | >> where $S_{f}$, $S_{fe}$, $k_1$, $k_2$ are material constants. In particular, $S_f$ is conceived as $f(T)$. | ||
| + | >>Additionally, $\mathcal{Y(k_{2}\cdot \underline{\sigma})}$ is a scalar function described as: | ||
| + | >>> $\mathcal{Y(\underline{\sigma})} = \frac{1+\nu}{2E}\Bigl[ \frac{\langle \underline{\sigma}_{ij} \rangle^{+} :\langle \underline{\sigma}_{ij} \rangle^{+}}{ (1-D)^2 } + h_{\text{mD}}\frac{\langle \underline{\sigma}_{ij} \rangle^{-} :\langle \underline{\sigma}_{ij} \rangle^{-}}{ (1-h_{\text{mD}}D)^2 } \Bigr] - \frac{\nu}{2E}\Bigl[ \frac{\langle \underline{\sigma}_{kk} \rangle^{2} }{ (1-D)^2 } + h_{\text{mD}}\frac{\langle \underline{\sigma}_{kk} \rangle^{2} }{ (1-h_{\text{mD}}D)^2 } \Bigr] $ \\ | ||
| + | >> __Creep damage: Kachanov law__ | ||
| + | >> 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}} $ | ||
| + | >> 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 ==== | ||
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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.1731664695.txt.gz · Last modified: 2024/11/15 10:58 by carlos
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