# Effect of the Computational Model and Mesh Strategy on the Springback Prediction of the Sandwich Material

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Static Tensile Test

_{p0.2}, ultimate strength R

_{m}, total ductility A

_{80mm}, uniform ductility A

_{g}and Young´s modulus E). The plastic strain ratio (Lankford parameter) r was determined in accordance with the standard ISO 10113 in the interval of true plastic strain values from 10% to 20%. The results of the individual material quantities are summarised in Table 2. Engineering stress R (MPa) vs. engineering strain ε (1), thus so-called engineering stress–strain curves (regarding the individual rolling directions) are shown in Figure 3. The evaluation of the mechanical values given in Table 2 was carried out by considering the measured force values to the whole cross-section, i.e., the measured values represent the average values of mechanical properties for the three layers of the tested sandwich material (steel-plastic-steel).

_{pl}+ ε

_{0})

^{n}

- σ—true stress (effective stress) (MPa)
- K—strength coefficient (MPa)
- ε
_{pl}—true plastic strain (1) - ε
_{0}—offset true strain (pre-strain) (1) - n—strain hardening exponent (1)

_{g}in the software OriginPro 9.0. The advantage of defining the hardening curve by using the Krupkowski hardening law is the simple possibility of integrating this function and thus it is easy to express the equivalent plastic work parametrically in dependence on the true strain and the approximation constants arising from such a function. An example of the approximation result can be seen in Figure 4. The results of the obtained approximation constants are presented in Table 3.

#### 2.2. Hydraulic Bulge Test

_{eff}(arising from the Tresca’s plastic flow criterion) can be finally expressed by relation (3).

- p—hydraulic pressure (MPa)
- R—radius of curvature (mm)
- t—actual thickness of specimen (mm)

_{eff}can be calculated using principal strains ε

_{1,2,3}according to Equation (4)

- ε
_{1,2,3}—principal strains (1)

_{3}(sometimes termed also as a thickness reduction) firstly via the constant volume law (5) and subsequently using Equation (6).

- t—actual thickness (mm)
- t
_{0}—initial thickness (mm)

_{1}, ε

_{2}), as well as the radius of curvature R must be detected to subsequently determine the stress–strain curve. For this purpose, two surfaces have been defined on the measured area, in which the system calculates the desired quantities. An example of the analysis area is shown in Figure 7. The larger analysis area (area A) is used to find the radius of curvature R and the smaller area (area B) is used to calculate the major and minor strain, from which the actual thickness of the material can be calculated using Equations (5) and (6). An example of true strain distribution in the relevant area and evolution of the radius of curvature R is shown in Figure 8.

#### 2.3. Cyclic Test

#### 2.4. U-Bending of the Specimens

## 3. Numerical Simulation

#### 3.1. Vegter Yield Criterion

_{1}and σ

_{2}. The angle θ denotes the relative rotation between the principal stresses and chosen coordinate system. The principle of the element planar loading and the individual stress components can be subsequently expressed by Equations (7) and (8).

_{PS}= 0.5 is very often assumed when defining the yield criterion. A similar problem is arising during the experimental determination of the point SH position characterising the pure shear stress. In this case, the presumption about the symmetry of yield criterion and parameter α

_{SH}having the value equal to 0.5 is used, as in the case of plane strain point. In the case of experimental determination of these points, no estimation of such extra input parameter α is necessary and the Vegter yield surface directly passes through these experimentally detected points.

_{1}and σ

_{2}the plane stress state condition (σ

_{3}= 0). The yield criterion is completely dependent on the three parameters mentioned above but is not uniquely determined due to the interdependence of the parabolic functions. The yield surface is described as the individual parts of the Bezier curve affected by the hinge-point B and passing through the reference points A and C. The total yield surface is constructed through the four quadrants expressing the different stress states. Continuity of the first derivative is required between any part of the yield function. The strain vector is derived as follows:

- s = sin (2θ)
- c = cos (2θ)

_{1}and σ

_{2}) and angle θ between the material axes and directions of the principal stresses.

#### 3.2. Kinematic Hardening Law

#### 3.3. Definition of the Material Model in the Software PAM-STAMP 2G

#### 3.3.1. Vegter Yield Criterion in Combination with the Isotropic Hardening Law

- x—weighted average of the monitored quantity
- x
_{0}, x_{45}, x_{90}—measured values in the relevant directions

#### 3.3.2. Vegter Yield Criterion in Combination with the Kinematic Hardening Law

#### 3.3.3. Vegter Yield Criterion in Combination with the Kinematic Hardening Law: Volume Element of the Deformation Mesh

## 4. Results from the Finite Element Analysis (FEA)

## 5. Discussion

## 6. Conclusions

- The isotropic hardening law cannot be used to correctly predict the springback of sandwich material in cases where the stress state changes during the forming process.
- The kinematic hardening law provides a more accurate springback prediction compared to the isotropic hardening model regardless of the surface or volume element selection for the computational mesh.
- The choice of the meshing strategy does not have any significant effect on the FEA result when the kinematic hardening law is used. The surface and volume elements give almost exactly comparable results for the springback prediction of the sandwich material.
- From a quantitative point of view (using histograms of surfaces’ deviations, see Figure 26, Figure 27 and Figure 28), it was confirmed that the kinematic hardening law (regardless of the element type) has significantly higher accuracy in springback prediction than the isotropic hardening law. In addition to that, both kinematic hardening laws (surface and volume type of mesh) have almost 90% of the surfaces’ deviations up to 0.5 mm compared to the isotropic hardening law, where only 46% can be found up to 0.5 mm.
- The definition of the sandwich material using layers of volume elements in the deformation mesh does not provide a significant improvement of the FEM result.
- In the numerical simulation of forming the sandwich material, the measured values of the mechanical quantities can be related to the entire sheet (sandwich) thickness, and it is not necessary to distinguish the different deformation and stress behaviour of the individual layers.
- From the calculation accuracy point of view, it does not make any sense to use volume elements of the deformation mesh for the thin sheets. Such an approach leads only to a significant increase in computational time.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**Layout of the cyclic test (

**left**) and results of the cyclic test for tested sandwich material (

**right**).

**Figure 15.**Definition of the plane strain point PS via extra input parameter α [50].

**Figure 16.**Change of mechanical properties due to the Bauschinger effect [51].

**Figure 17.**Material card (software PAM-STAMP 2G): Vegter yield criterion and isotropic hardening law.

**Figure 26.**Distribution of deviations between determined surfaces: Vegter–isotropic hardening law–surface elements.

**Figure 27.**Distribution of deviations between determined surfaces: Vegter–kinematic hardening law–surface elements.

**Figure 28.**Distribution of deviations between determined surfaces: Vegter–kinematic hardening law–volume elements.

Chemical Element | C | Si | Mn | Al | P | S | Ti | Nb |
---|---|---|---|---|---|---|---|---|

Composition [wt%] | 0.097 | 0.406 | 0.746 | 0.211 | 0.016 | 0.013 | 0.122 | 0.074 |

Direction | R_{p0.2} (MPa) | R_{m} (MPa) | A_{g} (%) | A_{80mm} (%) | r_{10–20} (1) | E (MPa) |
---|---|---|---|---|---|---|

0° | 291.7 | 426.5 | 17.8 | 23.1 | 1.335 | 174,283 |

45° | 299.6 | 418.3 | 18.9 | 27.1 | 1.515 | 183,718 |

90° | 307.7 | 428.5 | 18.1 | 26.6 | 1.622 | 179,928 |

Direction | K (MPa) | n (1) | ε_{0} (1) |
---|---|---|---|

0° | 720.6 | 0.1995 | 0.0030 |

45° | 707.6 | 0.2025 | 0.0065 |

90° | 726.9 | 0.2039 | 0.0067 |

Material | K (MPa) | n (1) | ε_{0} (1) |
---|---|---|---|

Sandwich | 865.1 | 0.267 | 0.0076 |

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**MDPI and ACS Style**

Solfronk, P.; Sobotka, J.; Koreček, D.
Effect of the Computational Model and Mesh Strategy on the Springback Prediction of the Sandwich Material. *Machines* **2022**, *10*, 114.
https://doi.org/10.3390/machines10020114

**AMA Style**

Solfronk P, Sobotka J, Koreček D.
Effect of the Computational Model and Mesh Strategy on the Springback Prediction of the Sandwich Material. *Machines*. 2022; 10(2):114.
https://doi.org/10.3390/machines10020114

**Chicago/Turabian Style**

Solfronk, Pavel, Jiří Sobotka, and David Koreček.
2022. "Effect of the Computational Model and Mesh Strategy on the Springback Prediction of the Sandwich Material" *Machines* 10, no. 2: 114.
https://doi.org/10.3390/machines10020114