Anisotropic Plasticity During Non-proportional Loading

Abstract

Modeling of the elasto-plastic behavior of isotropic and anisotropic metals for applications to forming process simulations is discussed. In particular, the macroscopic flow theory of plasticity combined with the concept of isotropic hardening, in which a single monotonic stress-strain curve serves as a reference, is briefly reviewed. Selected non-proportional loading test procedures are described and the main deviations of the material behavior compared to an isotropic hardening response are discussed on the basis of underlying mechanisms of deformation at lower scale. The failure of isotropic hardening to accurately capture the behavior of a material subjected to non-linear strain paths demonstrates the need for more advanced hardening theories. Thus, theories based on kinematic hardening, possibly combined with distortional plasticity concepts, are succinctly reviewed. A pressure-dependent, distortional-only, plasticity approach recently proposed is discussed in more details and its relevance is illustrated with the prediction of stress-strain curves of advanced high strength steel sheets deformed along non-linear strain paths. A finite element (FE) implementation of this distortional plasticity model is outlined, with special attention to the formulation of the stress integration algorithm and the elasto-plastic tangent tensor. Application examples on several steel sheet samples subjected to various strain path changes are given for validation purpose. Simulations are conducted with a stand-alone (SA) code containing the constitutive equations only and a FE code with only one element. Comparisons between these predictions and experimental results demonstrate the accuracy of the model and the excellent performance of the FE implementation. Applications on advanced high strength steel (AHSS) sheet demonstrate why the pressure-dependency in the model is an important feature.

Appendix

State Variable Evolution Equations

The yield condition is given above by combining Eq. (2.37) with (2.38) or (2.39). The development of the state variable evolution equations is explained in the original article (Barlat et al. 2020), and are only listed below.

• 1. Transformed deviators: A number of transformed stress deviators are introduced to describe cross-loading effects

  $$\begin{aligned} \textbf{s}_{C}=(\boldsymbol{\upsigma }^\prime \boldsymbol{:} \hat{\textbf{h}}) \hat{\textbf{h}} \end{aligned}$$

  (2.59)
  $$\begin{aligned} \textbf{s}_{O}=\boldsymbol{\upsigma }^\prime -\textbf{s}_{C} \end{aligned}$$

  (2.60)
  $$\begin{aligned} \textbf{s}_{X}=\xi _{C}\left( \frac{1-g_{C}}{g_{L}}\right) \textbf{s}_{O} \end{aligned}$$

  (2.61)
  $$\begin{aligned} \textbf{s}_{L}=\frac{1}{\xi _{L}\left( g_{L}-1\right) +1} \textbf{s}_{C}+\frac{1}{g_{L}} \textbf{s}_{O} \end{aligned}$$

  (2.62)

• 2. Cross-loading-modified effective stress: The captioned effective stress is defined as

  $$\begin{aligned} \bar{\xi }(\boldsymbol{\upsigma }^\prime )=\left[ \bar{\phi }\left( \textbf{s}_{L}\right) ^{p}+\bar{\phi }\left( \textbf{s}_{X}\right) ^{p}\right] ^{1 / p} \end{aligned}$$

  (2.63)

• 3. Strain path change amplitude: These parameters correspond to the severity of strain path change, i.e., 1 for monotonic, 0 for pure cross-loading and –1 for reverse loading

  $$\begin{aligned} \cos \chi =\hat{\textbf{h}}: \hat{\boldsymbol{\upsigma }}^\prime \;\;\;\;\cos \chi ^\prime =\hat{\textbf{h}^\prime }: \hat{\boldsymbol{\upsigma }}^\prime \end{aligned}$$

  (2.64)

• 4. Microstructure deviators: These are the tensors, both set equal to the stress deviator at the first step, which account for a smooth evolution of the microstructure

  $$\begin{aligned} \frac{d \hat{\textbf{h}}^{\prime }}{d \bar{\varepsilon }}=\lambda ^{\prime } k^{\prime }\left( \hat{\boldsymbol{\upsigma }}^\prime -\cos \chi ^{\prime } \hat{\textbf{h}}^{\prime }\right) \end{aligned}$$

  (2.65)
  $$\begin{aligned} \frac{d \hat{\textbf{h}}}{d \bar{\varepsilon }}=\lambda k\left( \cos \chi ^{\prime 2 \xi _{R}}+\cos \chi ^{2 \xi _{R}}\right) (\hat{\boldsymbol{\upsigma }}^\prime -\cos \chi \hat{\textbf{h}}) \end{aligned}$$

  (2.66)

• 5. Reverse loading: Reverse loading is described in the yield condition by the two state variables, $f_-$ and $f_+$, which are function of two other variables $g_-$ and $g_+$

  $$\begin{aligned} f_{\omega }=\sqrt{\frac{3}{8}}\left\{ \frac{1}{g_{\omega }^{q}}-1\right\} ^{\frac{1}{q}} \end{aligned}$$

  (2.67)

  where $\omega$ stands for the symbols − or $+$. The evolution equations are expressed with the variables $g_-$ and $g_+$

  $$\begin{aligned} \begin{aligned} \frac{d g_{-}}{d \bar{\varepsilon }}&=\frac{1+\lambda }{2}\left\{ k_{1} \frac{1-g_{-}}{g_{-}^{\xi _{B}^{\prime }}}\left[ 1-\left( \cos ^{2} \chi \right) ^{\xi _{B}}\right] \right. \\&\left. +k_{2}\left( k_{3} \frac{\sigma _{\textrm{y}}}{\sigma (\bar{\varepsilon })}-g_{-}\right) \left( \cos ^{2} \chi \right) ^{\xi _{B}} \right\} +\frac{1-\lambda }{2}\left\{ k_{1} \frac{1-g_{-}}{g_{-}^{\xi _{B}^{\prime }}}\right\} \end{aligned} \end{aligned}$$

  (2.68)
  $$\begin{aligned} \begin{aligned} \frac{d g_{+}}{d \bar{\varepsilon }}&=\frac{1-\lambda }{2}\left\{ k_{1} \frac{1-g_{+}}{g_{+}^{\xi _{B}^{\prime }}}\left[ 1-\left( \cos ^{2} \chi \right) ^{\xi _{B}}\right] \right. \\&\left. +k_{2}\left( k_{3} \frac{\sigma _{\textrm{y}}}{\sigma (\bar{\varepsilon })}-g_{+}\right) \left( \cos ^{2} \chi \right) ^{\xi _{B}}\right\} +\frac{1+\lambda }{2}\left\{ k_{1} \frac{1-g_{+}}{g_{+}^{\xi _{B}^{\prime }}}\right\} \end{aligned} \end{aligned}$$

  (2.69)

• 6. Permanent softening: Permanent softening should be maximum for strain reversals

  $$\begin{aligned} g_{P}^{\prime }=g_{P}^{*}-\left( g_{P}^{*}-g_{3}^{*}\right) \left| \hat{\boldsymbol{\upsigma }}^\prime \boldsymbol{:} \hat{\textbf{h}}^{*}\right| ^{2 \xi _{S}} \end{aligned}$$

  (2.70)
  $$\begin{aligned} \frac{d g_{3}}{d \bar{\varepsilon }}=k_{5} g_{S}\left( k_{4}-g_{3}\right) \end{aligned}$$

  (2.71)
  $$\begin{aligned} \frac{d g_{S}}{d \bar{\varepsilon }}=-k_{S}\left[ 1-\left| \hat{\boldsymbol{\upsigma }}^\prime \boldsymbol{:} \hat{\textbf{h}}^{*}\right| \right] g_{S} \end{aligned}$$

  (2.72)

• 7. Cross-loading contraction: The contraction and recovery of the yield surface are controlled by the next equations, respectively

  $$\begin{aligned} \frac{d g_{C}}{d \bar{\varepsilon }}=k_{C}\left( C-g_{C}\right) \end{aligned}$$

  (2.73)
  $$\begin{aligned} \frac{d g_{C}}{d \bar{\varepsilon }}=k_{C}^{\prime } \frac{1-g_{C}}{g_{C}^{\xi _{C}}} \end{aligned}$$

  (2.74)

• 8. Latent hardening: Finally, latent hardening is defined by

  $$\begin{aligned} \frac{d g_{L}}{d \bar{\varepsilon }}=k_{L}\left[ \frac{\sigma _{r}(\bar{\varepsilon })-\sigma _{r}(0)}{\sigma _{r}(\bar{\varepsilon })}\left( \sqrt{L+(1-L) \cos ^2 \chi ^{\prime \xi _{L}^{\prime }}}-1\right) +1-g_{L}\right] \end{aligned}$$

  (2.75)

Finite Difference Method

The first and second-order numerical derivatives of yield function are acquired based on the finite difference method (FDM) in a scaled stress space (Aretz 2007). The main advantages of the numerical derivatives for FE-implementation are that (1) the challenges for FE-implementation of advanced plasticity models are alleviated, (2) the singularities that appear in analytical derivatives do not require a special treatment, and (3) FDM is applicable to shell and solid elements corresponding to plane and 3D stress states, respectively. The scaled (normalized) stress space expands the utility of the FDM step size ($\updelta \hat{\upsigma }$) for various stress scales. As remarked in Choi and Yoon (2019), the midpoint rule (central difference method) accelerates the algorithm convergence and higher order of truncated error.

For the sake of the brevity, in this context, the stress tensor is expressed in the form of Voigt’s notation (vectorized form).

$$\begin{aligned} \left[ \boldsymbol{\upsigma }\right] =\left[ \begin{array}{lll} \sigma _{11} &{} \sigma _{12} &{} \sigma _{13} \\ \sigma _{12} &{} \sigma _{22} &{} \sigma _{23} \\ \sigma _{13} &{} \sigma _{23} &{} \sigma _{33} \end{array}\right] =\left\{ \begin{array}{c} \sigma _{11} \\ \sigma _{22} \\ \sigma _{33} \\ \sigma _{23} \\ \sigma _{13} \\ \sigma _{12} \end{array}\right\} =\left\{ \begin{array}{l} \sigma _{1} \\ \sigma _{2} \\ \sigma _{3} \\ \sigma _{4} \\ \sigma _{5} \\ \sigma _{6} \end{array}\right\} \end{aligned}$$

(2.76)

A stress tensor is scaled by the normalization factor ($\xi$).

$$\begin{aligned} \hat{\boldsymbol{\upsigma }}=\xi \boldsymbol{\upsigma }=\frac{\boldsymbol{\upsigma }}{\sqrt{\boldsymbol{\upsigma }: \boldsymbol{\upsigma }}} \end{aligned}$$

(2.77)

The first-order derivative is obtained based on a midpoint rule

$$\begin{aligned} \frac{\partial \bar{\sigma }(\hat{\boldsymbol{\upsigma }})}{\partial \hat{\sigma }_{i}} \cong \frac{\bar{\sigma }\left( \hat{\sigma }_{i}+\delta \hat{\sigma }, \hat{\sigma }_{j}\right) -\bar{\sigma }\left( \hat{\sigma }_{i}-\delta \hat{\sigma }, \hat{\sigma }_{j}\right) }{2(\delta \hat{\sigma })} \end{aligned}$$

(2.78)

where $i\ne {j}$ and $\{i, j\} \in \{1,2,3,4,5,6\}$. It is worth noting that the step size for the finite difference method ($\delta \hat{\sigma }$) is assumed to be constant for all components. $\delta \hat{\sigma }=10^{-6}$ is recommended for the first-order derivative. The normalization factor does not influence the values of the first derivative because the yield function derivative is a homogeneous function of first degree.

$$\begin{aligned} \frac{\partial \bar{\sigma }(\hat{\boldsymbol{\upsigma }})}{\partial \hat{\boldsymbol{\upsigma }}}=\frac{\partial \xi \bar{\sigma }(\boldsymbol{\upsigma })}{\partial \xi \boldsymbol{\upsigma }}=\frac{\partial \bar{\sigma }(\boldsymbol{\upsigma })}{\partial \boldsymbol{\upsigma }} \end{aligned}$$

(2.79)

The second-order derivative is defined differently based on the components, i.e., (1) diagonal ($i=j$) and (2) off-diagonal ($i\ne {j}$) and symmetry according to Schwarz’ theorem.

$$\begin{aligned} \begin{aligned}&\frac{\partial ^{2} \bar{\sigma }(\hat{\boldsymbol{\upsigma }})}{\partial \hat{\sigma }_{i} \partial \hat{\sigma }_{j}}=\frac{\bar{\sigma }_{+}-2 \bar{\sigma }(\hat{\boldsymbol{\upsigma }})+\bar{\sigma }_{-}}{(\delta \hat{\sigma })^{2}}&\boldsymbol{\textrm{IF}} \quad i=j \\&\frac{\partial ^{2} \bar{\upsigma }(\hat{\boldsymbol{\upsigma }})}{\partial \hat{\upsigma }_{\textrm{i}} \partial \hat{\upsigma }_{\textrm{j}}}=\frac{\bar{\upsigma }_{++}-\bar{\upsigma }_{+-}-\bar{\upsigma }_{-+}+\bar{\upsigma }_{--}}{(\delta \hat{\upsigma })^{2}}&\boldsymbol{\textrm{ELSE}} \end{aligned} \end{aligned}$$

(2.80)