#
Quadratic Midpoint Integration Method for J_{2} Metal Plasticity

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## Abstract

**:**

_{2}von Mises metal plasticity model are examined for the accuracy of deviatoric stress integration of the constitutive equation. The accuracy of stress integration using a strain rate vector for arbitrary direction is presented in terms of an iso-error map for comparison with the exact solution. Accuracy and stability issues of the quadratic integration method are discussed using a two-dimensional metal panel problem with a single slit-like defect in the center. The scale factor and shape factor were introduced to a quadratic integration rule for assuming a returning directional tensor from a trial stress onto the final yield surface. Luckily enough, the perfectly plastic model is the only case where the analytical solution is possible. Thus, solution accuracies were compared with those of the exact solutions. Since the standard scale factor ranges from 0 to 1, which is similar to the linear $\alpha $-method, the penalty scale factors that are greater than 1 were mainly explored to examine the solution accuracies and computational efficiency. A higher value of scale factor above five shows a better computational efficiency but a decreased solution accuracy, especially in the higher plastification zone. A well-balanced scale factor for both computational efficiency and solution accuracy was found to be between one and five. The trade-off scale factor was proposed to be five. The proper shape factor was also proposed to be {1,1,4}/6 among the different combinations of weight distribution over a time interval. This proposed scale factor and shape factor is also valid for relatively long time periods.

## 1. Introduction

_{2}metal plasticity. In addition, Krieg et al. [2] explored the accuracies of several solution methods, including the tangent stiffness, secant stiffness, and radial return method compared to the exact solutions obtained from the elastic perfectly plastic model. This perfectly plastic model is the only case where an analytical solution is possible. The exact solution was proposed by Krieg et al. [2] to the perfect plasticity equations for the general deviatoric stress increment. There were no approximations involved in deriving the starting differential equation of incremental plasticity. Based on of these pioneering papers, Ortiz and Popov [3] and Simo et al. [4,5] elaborated the return mapping algorithms for the plane stress problem. The linear $\alpha $ method was used to integrate the new stress state over the time interval. By assuming the new stress state in terms of a linear combination between current known stress and stress to be determined, the plastic corrector term has changed accordingly. This means that the direction of mapping onto the final yield surface has changed with varying $\alpha $ value. Thus, this directional change onto the yield surface may subsequently affect to computation time to calculate a numerical subroutine. To elaborate this issue, Willam [6] and Shim [7] proposed the higher order $\alpha $ method by expanding to additional time integration points and weights, while the time interval remained as a single step. Artioli et al. [8] and Jahanshahi [9] introduced multi-step methods for von Mises materials, specifically single-step midpoint method (SMPT1 and SMPT2) and double-step midpoint method (DMPT1 and DMPT2). The first two schemes used a generalized midpoint integration rule with return mapping procedure enforced by yield conditions. The latter two schemes were two-stage algorithms developed by dividing each time step into two subintervals, in which equations were solved in turn. These approaches essentially rely on sub-steps in a time interval. They develop an extensive comparison based on pointwise mixed stress-strain loading histories, iso-error maps and initial boundary value problem. They investigated the existence of solution, accuracy, stability and the algorithm behavior for long time steps. Since Willam [6] and Shim [7] proposed the higher order $\alpha $ method to evaluate the new stress state, they did not implement this scheme in a finite element problem. Hence, this paper examines the accuracies of the quadratic $\alpha $ method with the return mapping algorithm for solution of a uniaxial tension problem with a slit in the center. It is instructional to examine the solution using the perfectly plastic model, which only has the analytical solution available.

## 2. J_{2} Metal Plasticity

_{2}. This von Mises yield criterion is known to be in better agreement with experiments for ductile metals such as copper, nickel, aluminum, and alloy steels [12]. This paper focuses on the analytical formulation and numerical implementation of the von Mises criterion, which we shall simply refer to as J

_{2}plasticity. In metals, irreversible deformation is largely due to crystal dislocation, generally evident as irreversible deformation behavior when an applied load is removed.

**:**denotes an inner product (double contraction), and $I$ denotes the rank-four symmetric identity tensor. J

_{2}is the second invariants of the deviatoric stress tensor, $s$ is defined as

_{2}plasticity. The typical procedure of this algorithm is to write the stress tensor in the elastic predictor and plastic corrector format as in Equation (12).

## 3. J_{2} Metal Plasticity with Linear α Method

## 4. Accuracy of Quadratic Midpoint Integration Method

## 5. Implementation for Quadratic α Method

## 6. 2D Metal Panel Problem with Slit-Like Defect

## 7. Conclusions

_{2}von Mises metal plasticity model are examined for the accuracy of deviatoric stress integration of the constitutive equation. Based on the Lagrangian interpolation expansion for the linear $\alpha $ method, the scale factor and shape factor were introduced to a quadratic integration rule for assuming a returning directional tensor from a trial stress onto the final yield surface. As the perfectly plastic model is the only case where the analytical solution is possible, solution accuracies can be compared with those of the exact solutions with the aid of an iso-error map. Since the standard scale factor ranges from 0 to 1, which is similar to the linear $\alpha $-method, the penalty scale factors greater than 1 were mainly explored to examine the solution accuracies and computational efficiency. A higher value of scale factor above five shows a better computational efficiency but a decreased solution accuracy, especially in the higher plastification zone. A well-balanced scale factor for both computational efficiency and solution accuracy was found to be between one and five. The trade-off scale factor was proposed to be five. The proper shape factor was also proposed to {1,1,4}/6 among the different combinations of weight distribution over a time interval. Since the backward implicit scheme, $\alpha =1$ in the linear method shows the best performance in terms of solution accuracy and a computation time. In this quadratic $\alpha $-method, we tried to explore a penalty scale factor that is higher than 1. Although this approach can be seen in the non-logical sense at a glance, it shows better results in either solution accuracy or computation time. A smaller stress increment produced more severe positive solution errors mainly localized in the narrow region of the small tangential stress increment. For this reason, the backward Euler method in the linear $\alpha $-method still showed better performance. However, for large increments of stress such as the loading step (30, 12, 8 and 5 steps), the backward Euler method lost its stability under the limit of 300 iterations for seeking global Newton–Rapson loops. In contrast, the quadratic time integration scheme with 30, 12, 8 and 5 loading steps showed a stable solution. In addition, the quadratic SMPT2 scheme had a slightly higher efficiency compared to SMPT1.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Failure surface of J

_{2}plasticity: (

**a**) principal stress space, (

**b**) deviatoric stress space.

**Figure 3.**Deviatoric principal stress plane: (

**a**) radial and tangential coordinates at the contact stress, ${s}_{c}={s}_{n}$, and (

**b**) positive angular error $\theta $ shown between computed and exact stress is identified at each trial stress state, ${s}_{n+1}^{tr}$. ${s}_{n+1}$ is the final stress which was corrected by the radial return method. Note: r is the distance proportion between ${s}_{n}$ and ${s}_{c}$ over the time interval.

**Figure 4.**Radial return algorithm: trial stress over the yield surface and corrected final stress onto the yield surface: calculation and visualization using Mathematica.

**Figure 5.**Validation of the linear α-method: (

**a**) central difference, $\alpha =1/2$; (

**b**) skewed difference,$\alpha =2/3$; and (

**c**) backward Euler integration, $\alpha =1$.

**Figure 11.**Quadratic single-step midpoint methods (SMPT1 and SMPT2): standard α-method and penalty α-method.

**Figure 12.**Two-dimensional metal panel problem with slit-like defect: (

**a**) elements monitored, (

**b**) finite element mesh layout and (

**c**) von Mises stress contour plot.

**Figure 13.**Computational efficiency of Quadratic SMPT1 using the standard α-method and penalty α-method : the influences of different (

**a**) weights, (

**b**) values, (

**c**) which is the third component of, and (

**d**) time intervals.

**Figure 14.**Validation for quadratic α-method using von Mises stress at the element #837: (

**a**) varies from 1 to 100, while is kept constant as {0,1,0}, (

**b**) time steps varies from 5 to 30, while and are kept constants as 5 and {1,1,4}/6, respectively.

Time Integration Schemes | 30 Steps | 12 Steps | 8 Steps | 5 Steps | ||
---|---|---|---|---|---|---|

Quadratic $\alpha $-SMPT1 | All $\alpha $ | $\mathsf{\beta}=\{0,0,1\}$ | 2578.4 | 1019.6 | 660.8 | 392.1 |

$\alpha =1$ | $\mathsf{\beta}=\{0,1,0\}$ | 2377.6 | 998.3 | 648.2 | 385.4 | |

$\alpha =2$ | 2235.1 | 846.8 | 549.3 | 323.2 | ||

$\alpha =3$ | 1925.9 | 781.5 | 488.2 | 305.3 | ||

$\alpha =5$ | 1823.1 | 678.5 | 436.0 | 285.0 | ||

$\alpha =10$ | 1533.0 | 575.7 | diverged ^{1} | 263.1 | ||

$\alpha =35$ | 1082.1 | 530.3 | 367.3 | diverged ^{1} | ||

$\alpha =5$ | $\mathsf{\beta}=(1/6)\times \{1,4,1\}$ | 1984.4 | 754.3 | 479.4 | 303.2 | |

$\mathsf{\beta}=(1/6)\times \{4,1,1\}$ | 2521.9 | 918.4 | 651.5 | 387.0 | ||

$\mathsf{\beta}=(1/6)\times \{1,1,4\}$ | 2234.2 | 878.9 | 591.2 | 358.3 | ||

$\mathsf{\beta}=(2/6)\times \{1,1,1\}$ | 2228.9 | 860.5 | 549.2 | 326.4 | ||

$\alpha =35$ | $\mathsf{\beta}=(1/6)\times \{1,1,4\}$ | 1717.3 | 606.0 | 411.4 | 276.7 | |

$\mathsf{\beta}=(1/6)\times \{0,0,300\}$ | 1092.7 | 542.3 | diverged ^{1} | diverged ^{1} | ||

Quadratic $\alpha $-SMPT2 | $\alpha =1$ | $\mathsf{\beta}=\{0,1,0\}$ | 2523.1 | 992.8 | 640.3 | 378.1 |

$\alpha =2$ | 2256.8 | 868.4 | 552.4 | 319.2 | ||

$\alpha =3$ | 2055.7 | 753.2 | 486.2 | 303.0 | ||

$\alpha =5$ | 1833.7 | 682.1 | 422.1 | 266.7 | ||

$\alpha =10$ | 1541.8 | 543.1 | diverged ^{1} | diverged ^{1} | ||

$\alpha =35$ | 1137.9 | diverged ^{1} | diverged ^{1} | diverged ^{1} | ||

$\alpha =5$ | $\mathsf{\beta}=(1/6)\times \{1,4,1\}$ | 2011.6 | 693.8 | 476.6 | 297.4 | |

$\mathsf{\beta}=(1/6)\times \{4,1,1\}$ | 2540.2 | 990.2 | 638.5 | 378.7 | ||

$\mathsf{\beta}=(1/6)\times \{1,1,4\}$ | 2206.7 | 884.0 | 594.2 | 352.0 | ||

$\mathsf{\beta}=(2/6)\times \{1,1,1\}$ | 2285.7 | 858.0 | 552.0 | 319.2 | ||

$\alpha =35$ | $\mathsf{\beta}=(1/6)\times \{1,1,4\}$ | 1728.6 | 610.4 | diverged ^{1} | 270.4 | |

$\mathsf{\beta}=(1/6)\times \{0,0,300\}$ | diverged ^{1} | diverged ^{1} | diverged ^{1} | diverged ^{1} |

^{1}Solution had been exceeded 300 iterations for global Newton–Raphson iteration loop.

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**MDPI and ACS Style**

Rhee, I.; Kim, W.
Quadratic Midpoint Integration Method for J_{2} Metal Plasticity. *Metals* **2018**, *8*, 66.
https://doi.org/10.3390/met8010066

**AMA Style**

Rhee I, Kim W.
Quadratic Midpoint Integration Method for J_{2} Metal Plasticity. *Metals*. 2018; 8(1):66.
https://doi.org/10.3390/met8010066

**Chicago/Turabian Style**

Rhee, Inkyu, and Woo Kim.
2018. "Quadratic Midpoint Integration Method for J_{2} Metal Plasticity" *Metals* 8, no. 1: 66.
https://doi.org/10.3390/met8010066