Lagamine

Von Mises Visco-Plastic.

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.

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 (C.Rojas-Ulloa et al., 2024) for more details). This colaw is based in the work of Hélène Morch (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.

The yield surface is defined by the von-Mises yield criterion: an isotropic $J_{2}$-type function of the form:

$J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})=\Bigl[\frac{3}{2}(\underline{\tilde{S}}-\underline{\mathbb{X}}):(\underline{\tilde{S}}-\underline{\mathbb{X}})\Bigr]^{0.5}$

where $\underline{\tilde{\sigma}}$ is the effective stress tensor. It is calculated as function of the unitary damage $D$, $0 \leq D \leq 1$ as: $\underline{\tilde{\sigma}}=\underline{\sigma}\cdot (1-D)^{-1}$

$\underline{\tilde{S}}$ is the deviatoric stress tensor, calculated as:

$\underline{\tilde{S}} = \underline{\tilde{\sigma}}-\frac{1}{3}tr(\underline{\tilde{\sigma}})\underline{\mathbb{I}}$

The function $\Phi$ defining the yield criteiron is:
$\Phi = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})-\sigma_{y}-R \leq 0$

where:
* $\sigma_{y}$ is the yield stress of the material.
* $R$ is the isotorpic hardening, calculated as the Voce iso. hardening law:
$\dot{R} = -b\cdot (Q-R)\cdot \dot{p} \hspace{1cm} \rightarrow \hspace{1cm} R(p)=Q\cdot (1-\exp(-b\cdot p))$
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.
$\dot{\underline{\mathbb{X}}} = \displaystyle{\sum_{i=1}^{nAF}} \frac{2}{3}C_{i}\dot{\underline{p}} - \gamma_{i}\Bigl( \dot{\underline{\mathbb{X}}}_{i} - \dot{\underline{\mathbb{Y}}}_{i} \Bigr)\dot{p} - b_{i}J_{2}\Bigl( \underline{\mathbb{X}}_{i} \Bigr)^{r_{i}-1} \underline{\mathbb{X}}_{i} + \frac{1}{C_i}\frac{\partial C_{i}}{\partial T}\dot{T} \underline{\mathbb{X}}_{i} $

Within the $\sum$ term, from left to right:

The first term is a Swift-type kinematic hardening.
Here, $C_{i}$ is the only material constant, and $\dot{\underline{p}}$ is the plastic strain rate vector (Voigt notation in Lagamine).

The second term is intended to model the static recovery of the material.
Following the Chaboche formulation, it addresses terms for:

Mean stress evolution $\underline{\mathbb{Y}}$. Following Chaboche formulation and H. Morch's work, the mean-stress is conceived as the summation of a total of a user-defined number $j$ of terms $0 \leq j\leq i$. In its time-dependent variational form, the $j^{\text{th}}$ equation is calculated as:
$\dot{\underline{\mathbb{Y}}}_j=\alpha_{b_{j}}\cdot \Bigl( \frac{3}{2} Y_{\text{st}_{j}}\frac{\underline{\mathbb{X}}_{j}}{J_{2}(\underline{\mathbb{X}}_{j})} + \underline{\mathbb{Y}}_{j} \Bigr) J_{2}(\underline{\mathbb{X}}_{j})^{r_{j}-1} $
···
Strain memory surface $\gamma_{i}$. This Chaboche formulation is intended to model the non-masing behavior observed in certain Ni-based and martensitic alloys. Similarly, the total strain memory surface is the result of a sum of a user-defined number $k$ equations $k\geq 0$. The form of the $h^{\text{th}}$ equation in its variational (time-dependent) is:
$\dot{\gamma_{k}} = D_{\gamma_{k}}\Bigl( \gamma_{k}^{0} - \gamma_{k} \Bigr) \dot{p}$
where $D_{\gamma_{k}}$ is a material parameter, and the term $\gamma_{k}^{0}$ is a function of the form:
$\gamma_{k}^{0}=a_{\gamma_{k}} + b_{\gamma_{k}}\exp\bigl( -c_{\gamma_{k}}\text{q} \bigr)$
where $\text{q}$ is the norm of the equivalent plastic strain $p$ in the loading history $(\text{q} = \underline{p}:\underline{p})$.

The third term deals with the dynamic recovery of the material.
Here, $b_i$ and $r_i$ are user-defined material parameters.

The fourth term is made for non-isothermal plasticity effects.
This term is important in presence of thermal gradients $\dot{T}$, and if we introduce the parameter $C_i$ as function of temperature.

As a viscoplastiv law, VMVP includes a total of 3 viscoplastic functions.

option ivp=1: Norton law

This is the classical Norton constitutive law, described as:
$\dot{p} = \langle \frac{\sigma_v}{K} \rangle ^{n}$
where:
· $K$ and $n$ are the drag stress and Norton exponent (material parameters).
· $\sigma_{v}$ is the overstress, defined as the stress resulting from the calculation of the yield function $\Phi=0$, i.e.:
$\sigma_{v} = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}}) - \sigma_{y}-R$

option ivp=2: Graham-Walles creep equations

This set of equations are intended to describe the total creep response as a summation of a total of $nvp$ user-defined number of individual functions, each of them representing different hardening/softening creep regimes. Each $l$ function is of the form:
$\dot{p} = \displaystyle{\sum_{l=1}^{nvp}} \Bigl[ K_{l}\exp{\Bigl(\frac{T}{C_l}\Bigr)} J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}})^{n_l} p^{m_l} \Bigr] + |\dot{T}| K_{T} J_{2}(\underline{\sigma}) p^{m_T}$
where $K_{l}, C_{l}, n_{l}, m_{l}$ are user-defined material parameters for each $g$ creep equation, and $K_D, m_T$ are user-defined parameters for an additional creep-fatigue interaction term.

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:

Activation Function. The Activation Function is of the form:
$\exp \Bigl[ -a(1.05 + p - 0.001b)^{-c} \Bigr]$
where $a, b, c$ are material parameters. Strictly speaking however, these are temperature and stress dependent functions. With all experimental and theoretical information available so far, we have reached the following mathematical formulations for each parameter:

$a(\sigma_v,T) = \frac{a_1}{1+\exp \bigl( \frac{\sigma_v - p_2}{a_3} \bigr)}$
$b(\sigma_v,T) = \frac{x_1}{1+\exp \bigl( \frac{p_2 - \sigma_v}{b_3} \bigr)} + b_4$
$c(\sigma_v,T) = \frac{c_1}{1+\exp \bigl( \frac{\sigma_v - p_2}{c_3} \bigr)}$
where $a_1, a_3, b_1, b_3, b_4, c_1, c_3$ are material parameters, and $p_2$ is a temperature-dependent function defined as an Arrhenius-type function:
$p_{2}(T) = p_{21} \Big[ 1 - p_{22} \exp \Big( \frac{T}{p_{23}} \Big)\Big]$
where $p_{21}, p_{22}, p_{23}$ are user-defined material parameters.