/MAT/LAW87 (BARLAT2000)

Yield stresses can be defined either by user-defined functions (plastic strain versus stress) or analytically by a combination of Swift-Voce model. The model is based on Barlat YLD2000 criterion. ¹

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Definition

Comments

1. The yield function is expressed as:
   $$f={\sigma }_{eq}\left(\sigma \right)-{\sigma }_Y$$

   $\sigma$

   Cauchy stress tensor

   ${\sigma }_Y$

   Yield stress

   ${\sigma }_{eq}$

   Barlat 2000 equivalent stress computed as follows:

   Where,

   $${\sigma }_{eq}= \frac{1}{2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}}({\phi }^{\text{'}}\left(X^{\text{'}}\right)+{\phi }^{\text{'}\text{'}}\left(X^{\text{'}\text{'}}\right)){}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}$$
   $${\phi }^{\prime }\left(X^{\prime }\right)={\left|{X^{\prime }}_1-{X^{\prime }}_2\right|}^a$$
   $${\phi }^{″}\left(X^{″}\right)={\left|2{X^{″}}_2+{X^{″}}_1\right|}^a+{\left|2{X^{″}}_1+{X^{″}}_2\right|}^a$$

   $X\text{'}$ and $X"$ denote the principal values of the tensors $X\text{'}$ and $X"$ which are a linear transformation of the stress deviator, which leads to:

   $${\phi }^{\text{'}}\left(X^{\text{'}}\right)= {\left[{({X^{\text{'}}}_{xx}-{X^{\text{'}}}_{yy})}^2+4{\left({X^{\text{'}}}_{xy}\right)}^2\right]}^{\frac{a}{2}}$$
   $$\begin{array}{l} {\phi }^{\text{'}\text{'}}\left(X^{\text{'}\text{'}}\right)={\left[\frac{3}{2}({X^{\text{'}\text{'}}}_{xx}-{X^{\text{'}\text{'}}}_{yy})+\frac{1}{2}\sqrt{{({X^{\text{'}\text{'}}}_{xx}-{X^{\text{'}\text{'}}}_{yy})}^2+4{\left({X^{\text{'}\text{'}}}_{xy}\right)}^2}\right]}^a+\\ {\left[\frac{3}{2}({X^{\text{'}\text{'}}}_{xx}-{X^{\text{'}\text{'}}}_{yy})-\frac{1}{2}\sqrt{{({X^{\text{'}\text{'}}}_{xx}-{X^{\text{'}\text{'}}}_{yy})}^2+4{\left({X^{\text{'}\text{'}}}_{xy}\right)}^2}\right]}^a\end{array}$$

   The tensors $X\text{'}$ and $X"$ are linear transformations of the stress tensor:

   $X^{\prime }=L^{\prime }\sigma and X^{″}=L^{″}\sigma$

   $${\mathbf{L}}^{\mathbf{\text{'}}}=\frac{1}{3}\left[\begin{array}{ccc}2{\alpha }_1& -{\alpha }_1& 0\\ -{\alpha }_2& 2{\alpha }_2& 0\\ 0& 0& 3{\alpha }_7\end{array}\right]$$
   $${\mathbf{L}}^{\mathbf{\text{'}\text{'}}}=\frac{1}{9}\left[\begin{array}{ccc}-2{\alpha }_3+2{\alpha }_4+8{\alpha }_5-2{\alpha }_6& {\alpha }_3-4{\alpha }_4-4{\alpha }_5+4{\alpha }_6& 0\\ 4{\alpha }_3-4{\alpha }_4-4{\alpha }_5+{\alpha }_6& -2{\alpha }_3+8{\alpha }_4+2{\alpha }_5-2{\alpha }_6& 0\\ 0& 0& 9{\alpha }_8\end{array}\right]$$

2. The yield stress could be defined either by tabulated input or using the analytic Swift-Voce model.
   • Iflag=0: Tabulated yield stress vs plastic strain evolution.

     ${\sigma }_Y=f\left({\epsilon }_p\right)$

     It is possible to add strain rate dependency by defining a number N_rate of functions, one for each measured strain rate.

     ${\sigma }_Y=f\left({\epsilon }_p,\dot{\epsilon }\right)$

   • Iflag = 1: The analytic Swift-Voce model is expressed as:
     $${\sigma }_y={\alpha }_{sv}\left[A{\left({\epsilon }_p+{\epsilon }_0\right)}^n\right]+\left(1-{\alpha }_{sv}\right)\left[K_0+Q\left(1-\exp \left(-B{\epsilon }_p\right)\right)\right]$$

     Where, ${\epsilon }_p$ is the equivalent plastic strain.

   • Iflag=2: Hansel hardening model is considered.
     $${\sigma }_y=\left\{B_{HS}-\left(B_{HS}-A_{HS}\right)e^{\left(-m{\left[{\overline{\epsilon }}^p+{\epsilon }_0\right]}^n\right)}\right\}\left(K_1-K_{2}T\right)+\text{Δ}{H}_{\gamma \alpha }{V}_m$$

     Temperature is updated in the law when adiabatic conditions:

     $$\text{Δ}T=\eta \frac{{\sigma }_{eq}\Delta {\epsilon }_p}{\rho C_p}$$

     The martensite rate equation is computed as follows:

     $$\frac{\partial V_m}{\partial \epsilon }=\left\{\begin{array}{cc}0& if\epsilon <{\epsilon }_0\\ \frac{B}{A}\cdot e^{\left(\frac{Q}{T}\right)}\cdot {\left(\frac{1-V_m}{V_m}\right)}^{\left(\frac{B+1}{B}\right)}\cdot \frac{{\left(V_m\right)}^p}{2}\cdot \left(1-\tanh \left[C+DT\right]\right)& f\epsilon \ge {\epsilon }_0\end{array}\right.$$

   • Iflag = 3: 3 directions orthotropic tabulated yield stress formulation is considered:
     $${\sigma }_y=Q_1+Q_2\cos \left(2\theta \right)+Q_3\cos \left(4\theta \right)$$

     with $Q_1=\frac{{\sigma }_Y^0+2{\sigma }_Y^{45}+{\sigma }_Y^{90}}{4}$ , $Q_2=\frac{{\sigma }_Y^0-{\sigma }_Y^{90}}{2}$ , $Q_3=\frac{{\sigma }_Y^0-2{\sigma }_Y^{45}+{\sigma }_Y^{90}}{4}$

     Where, ${\sigma }_Y^{XX}$ is the yield stress in the XX-degree orthotropic direction, and $\theta$ is the current loading direction. For this formulation, the current yield stress is then interpolated using a Fourier series, considering the current loading direction. Each interpolation factor is computed according to the values taken by the yield stresses in the direction of 0, 45 and 90 degrees. Note that the 3 input tabulated yield stresses can be a 1-dimension table (yield stress versus plastic strain) or a 2-dimension table (yield stress versus plastic strain versus strain rate).

3. If Chard > 0 and Ikin =1, a kinematic hardening model of Chaboche Rousselier is used:

   The back stress is calculated, as follows:

   $$\mathit{\beta }=\sum _{i=1}^4{\mathit{\beta }}_{\mathit{i}}$$
   With,
   $$\text{Δ}{\mathit{\beta }}_{\mathit{i}}=Chard \left(A_i{C}_i\text{Δ}{\mathit{\epsilon }}_{\mathbf{p}}-C_i{\mathit{\beta }}_{\mathbf{i}}{\text{Δε}}_p\right)$$

   If Chard > 0 and Ikin =2, a kinematic hardening model of Prager is used. The backstress is calculated as follows:

   $$\text{Δ}\mathit{\beta }=Chard·c_0\text{Δ}{\mathit{\epsilon }}_{\mathbf{p}}$$

   Where, $c_0$ is a parameter that is automatically computed according to the isotropic hardening modulus. For the mixed isotropic-kinematic hardening combination, the yield stress is computed considering the value of the $Chard$ parameter:

   $${\sigma }_Y^{Mixed}=(1-Chard).{\sigma }_Y+Chard.{\sigma }_{Y0}$$

   Where, ${\sigma }_{Y0}$ is the initial yield stress for which ${\epsilon }_p=0$ .

   Note: Where kinematic hardening is activated, the yield function becomes

   $$f=\overline{\sigma }\left(\stackrel{=}{\mathit{\sigma }}-\stackrel{=}{\mathit{\beta }}\right)-{\sigma }_Y^{Mixed}$$
4. A strain rate filtering can be used to avoid noisy results.
   • If VP = 0 (dependency on total strain rate), it can be activated with the flag $F_{smooth}$ or by defining a $F_{cut}$ value. If no filtering frequency is defined, a default value 10kHz is set up.
   • If VP = 1 (dependency on plastic strain rate), the filtering is activated by default. The filtering frequency can be modified by you; otherwise, a default value of 10 kHz is used.
5. When Iflag=1 (analytic Swift-Voce formulation) is used, strain rate dependency is modelled with Cowper-Symonds expression:
   $${\sigma }_y={\sigma }_y\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\right)$$

   If c=0 or p=0, the strain rate effects are not considered.

6. If I_fit=1, the coefficients ${\alpha }_i$ will be automatically fit in the Starter. The tensile yield strengths ${\sigma }_{00},{\sigma }_{45},{\sigma }_{90}$ and Lankford ratios $r_{00},r_{45},r_{90}$ must be determined from uniaxial tension experiments along the rolling, diagonal and transverse directions at an amount of plastic work corresponding to a plastic strain equal to 0.2%. ${\sigma }_b$ and $r_b$ should be determined from biaxial test, for the same amount of plastic strain.