# Data-Oriented Constitutive Modeling of Plasticity in Metals

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Anisotropic Continuum Plasticity

#### 2.2. Stress Space in Cylindrical Coordinates

#### 2.3. Data-Oriented Yield Function

## 3. Results

#### 3.1. Training of ML Yield Function

#### 3.2. Application of The Trained ML Yield Function in FE Analysis

#### 3.3. Tresca Flow Rule

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Yield locus for plane-stress conditions (${\sigma}_{3}=0$) and Hill-like anisotropy with parameters given in Table 1 (red line) and for an isotropic material with the same yield strength, but ${H}_{1}={H}_{2}={H}_{3}=1$ (blue line). The values of the principal stresses are normalized by the yield strength ${\sigma}_{y}$.

**Figure 2.**Polar plots of a subset of the training data produced from the anisotropic yield function of the reference material: (

**a**) Von Mises (J2) equivalent stresses according to Equation (3) are used, such that the yield strength, rather than the equivalent stress, is a function of the polar angle $\theta $. (

**b**) Equivalent stresses are calculated according to the Hill definition in Equation (5) to achieve a constant yield strength by mapping the equivalent stresses accordingly. In both figures, the yield locus is indicated by a solid black line, data points in the elastic regime are plotted in blue color and data in the plastic regime in red color. Both figures represent the same stress data, only mapped in a different way; all stresses are normalized with the reference yield strength ${\sigma}_{y}$.

**Figure 3.**Field plot of the trained SVM decision function defined in Equation (18), where areas in purple color shades represent negative values and brown shades represent positive values. The numerical value of the decision function is not relevant because only its sign is taken into account in the flow rule. The isoline for ${f}_{\mathrm{SVC}}=0$ is represented as a black line. Training data are plotted in light blue color for data with negative values (elastic) and in brown color for positive values (plastic).

**Figure 4.**Color map of the trained SVC prediction of the yield function in slices through the principal stress space defined by plane-stress conditions: (

**a**) ${\sigma}_{3}=0$; (

**b**) ${\sigma}_{1}=0$; (

**c**) ${\sigma}_{2}=0$. Brown regions indicate values of “+1” (plasticity) and purple regions values of “−1” (elasticity). The ML yield locus, corresponding to the isoline for ${f}_{\mathrm{SVC}}=0$, is represented as a black line; the yield locus of the Hill-like anisotropic reference material is indicated as a red line. The training data points are plotted with the same color code as in Figure 3.

**Figure 5.**The finite element model on which four different load cases are studied consists of four quadrilateral elements (green) with linear shape function, and at total of nine nodes (red) situated at the corners of the elements. The bottom and left-hand-side boundary nodes are restricted to zero normal displacement (blue triangles), and the loading is applied on top and right-hand-side-nodes (blue arrows), as described in the text.

**Figure 6.**Stress strain curves obtained for elastic-ideal plastic material behavior under the loading conditions specified in the legend: (

**a**) Equivalent total strain vs. equivalent Hill-stress, (

**b**) equivalent total strain vs. equivalent J2-stress for Hill-like yield function, and (

**c**) equivalent total strain vs. equivalent J2-stress for ML yield function.

**Figure 7.**Stress states obtained for the four different plane-stress load cases are plotted in the ${\sigma}_{1}$-${\sigma}_{2}$ plane together with the yield loci of the trained ML flow rule and the Hill-like reference material. The flow stresses resulting from the ML yield function are plotted as small yellow circles and those from anisotropic Hill plasticity as large blue circles.

**Figure 8.**Field plot of the ML yield function trained with data generated from a reference material with a Tresca yield criterion, where areas in purple color shades represent negative values and brown shades represent positive values. The isoline, where the ML yield function is zero, is plotted as black line. The test data points are plotted as brown circles, for stresses in the plastic regime, and a s light blue circles for stresses in the elastic regime.

**Figure 9.**Results of FEA on the material with an ML flow rule trained with data obtained from a Tresca yield criterion: (

**a**) Equivalent J2 stress plotted over the equivalent total strain, for the four different load cases given in Table 2. (

**b**) Flow stresses obtained with the ML yield function plotted as yellow circles in the ${\sigma}_{1}$-${\sigma}_{2}$ principle stress space, together with the yield loci of the ML flow rule, the Tresca flow rule and an isotropic J2 flow rule.

**Table 1.**Elastic and plastic material parameters defining the reference material with Hill-like anisotropy in its plastic flow behavior. For simplicity, isotropic elastic behavior and ideal plasticity without work hardening are assumed in this work.

Quantity | Symbol | Value |
---|---|---|

Young’s modulus | E | 200 GPa |

Poisson’s number | $\nu $ | 0.3 |

Yield strength | ${\sigma}_{y}$ | 150 MPa |

Hill parameters | ${H}_{1},{H}_{2},{H}_{3}$ | 0.7, 1, 1.4 |

**Table 2.**Yield stresses (YS) obtained for Hill-like yield function, with parameters given in Table 1, and machine learning (ML) yield function under the specified load cases. The relative errors in yield stress and equivalent plastic strain (PE) at maximum load are also specified.

Load Case | YS-Hill (MPa) | YS-ML (MPa) | Rel. Error YS | Rel. Error PE |
---|---|---|---|---|

uniaxial stress, horizontal | 146.4 | 148.4 | 1.41% | −1.95% |

uniaxial stress, vertical | 162.7 | 162.2 | −0.3% | −0.45% |

equibiaxial strain | 136.9 | 139.5 | 1.88% | −1.36% |

pure shear strain | 161.1 | 159.7 | −0.87% | 0.24% |

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**MDPI and ACS Style**

Hartmaier, A. Data-Oriented Constitutive Modeling of Plasticity in Metals. *Materials* **2020**, *13*, 1600.
https://doi.org/10.3390/ma13071600

**AMA Style**

Hartmaier A. Data-Oriented Constitutive Modeling of Plasticity in Metals. *Materials*. 2020; 13(7):1600.
https://doi.org/10.3390/ma13071600

**Chicago/Turabian Style**

Hartmaier, Alexander. 2020. "Data-Oriented Constitutive Modeling of Plasticity in Metals" *Materials* 13, no. 7: 1600.
https://doi.org/10.3390/ma13071600