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.:
$\sigma_{v} = J_{2}(\underline{\tilde{\sigma}}-\underline{\mathbb{X}}) - \sigma_{y}-R$