# A Semianalytical Model for the Determination of Bistability and Curvature of Metallic Cylindrical Shells

^{*}

## Abstract

**:**

_{1}= 6 mm and R

_{2}= 6 mm. With the aim of model verification, experiments with a closed-die incremental bending tool were performed. Calculated and experimental results show good correlation regarding bistability and curvature. In addition, X-ray diffraction measurement of the residual stresses shows a good qualitative agreement regarding the calculated and experimental results.

## 1. Introduction

- 1.
- First incremental die-bending of the shell around the Y-axis with bending radius R
_{1}. - 2.
- After springback, the shell is turned upside down and flattened by elastic bending.
- 3.
- Second incremental die-bending of the shell around the X-axis with bending radius R
_{2}.

_{1}= R

_{2}= 6 mm. The manufactured fully closed structure is shown in Figure 3 and has a radius of 40.95 mm at the deployed (fully closed) configuration and a radius of 100.82 mm at the folded configuration.

^{3}including 81 Simpsons integration points across the shell thickness. The contact conditions between shell and bending tools were described by a “surface-to-surface” approach. All tool geometries were considered to be rigid bodies.

_{1}= R

_{2}= 6 mm was identified as a promising concept. Despite the good qualitative agreement between simulation and experiment, the quantitative results show a significant difference in prediction of the stable state radii. There are also some considerable drawbacks in the FE model regarding simulation time. A change of the shell thickness or the material and the corresponding material data requires an adaption of the FE model. Moreover, an FE simulation takes about 20 min for calculation, which is too slow for parameter studies with a big amount of variables. Thus, a previous determination of a promising process sequence and corresponding bending radii for a wide range of materials and shell thickness combinations is costly.

## 2. Semianalytical Model

#### 2.1. Model Assumptions

- Plane strain condition due to the large ratio of shell width to shell thickness.
- No stresses acting normal to the shell surface.
- Application of pure bending load.
- No change of the shell thickness during bending.
- Neutral bending axis is always in the shell center.
- In tension and compression, the stress–strain characteristics of the material are the same.
- Planes perpendicular to the neutral axis stay perpendicular to this axis during bending.

#### 2.2. Model Description

_{1}. During plastic forming, kinematic hardening occurs and the yield locus is translated from the original position by an amount equal to the vector of plastic stress accumulated during plastic forming. After plastic forming, the material unloads linearly until point D, which corresponds to the springback configuration with the shell curvature of k

_{sb}. After springback, the shell is bended further until the flat state (point E). From the flattened state, the bending over the X-axis starts. This step also consists of elastic deformation until point F and plastic forming until point G. After the second bending, the shell has the curvature of k

_{2}. Unlike the first bending, plastic forming at the second bending operation leads to isotropic hardening [7], and the yield locus grows. Point H corresponds to the springback configuration after the second bending operation, which is equal to the first stable state with the shell curvature of k

_{1.geom,y}. By bending the shell to the flat geometry (point I) and further bending in the opposite direction over the Y-axis, the second stable state could be achieved at point J with the shell curvature of k

_{2.geom,x}. In the next sections, this bending sequence will be described in detail for each single step.

_{1,x}> 0). After bending the shell in reverse direction along the X-axis (k

_{2,y}< 0) and at the first stable state, the curvature has a negative sign (k

_{1.geom,y}< 0). At the second stable state, when the shell is bent along the Y-axis, the curvature again has a positive sign (k

_{2.geom,x}> 0). Taking this into account, an assessment of the stability of the shell geometry becomes possible. If the first and second geometry have an opposite sign in curvature other than defined, there is no stable geometry along the given axis, and, correspondingly, the shell is not bistable.

#### 2.2.1. Elastic Deformation during the First Bending

_{C}could be divided into two parts: elastic stresses and plastic stresses (Figure 5b). The starting point of the process is an elastic bending within the first bending operation. In general, stresses and strains are considered in two dimensions according to the above-mentioned assumptions. For means of simplification, a vector notation is used as given in Equation (1).

_{1}is the first bending curvature, v is the Poisson’s ratio, σ

_{0}is the initial yield stress of the material, and z gives the distance from midsurface of the shell. The maximum elastic strain and elastic strain increment follow accordingly and are given in Equations (3) and (4).

#### 2.2.2. Plastic Forming during the First Bending

_{t}, which gives the slope of the yield locus at a given stress state.

_{0}is the initial hardening rate, θ

_{1}is the asymptotic hardening rate, and (σ

_{0}+ σ

_{1}) is the back-extrapolated yield stress, with σ

_{0}as the initial yield stress. After plastic forming starts, the stresses acting on the material can be calculated with the help of the material tangent stiffness matrix as given in Equation (12) [7].

#### 2.2.3. First Elastic Springback

_{x}and m

_{y}are the resultant bending moments per unit length in the unit Nmm/mm, and σ

_{x}(z) and σ

_{y}(z) are the residual stress distributions along the given axes. In order to determine the remaining curvature after springback, the linear relationship between bending moment and resulting shell curvature [12] can be applied according to Equation (18) [13]:

_{E,x}) and bending moments along X-axis at springback curvature (m

_{D,x}) are equal to zero, the springback curvature can be calculated according to Equation (19).

#### 2.2.4. Elastic Bending in the Reverse Direction

_{1}should be subtracted from the stresses after first bending (Equation (20)):

_{F}), can be determined by solving a system of two equations (Equation (21)):

#### 2.2.5. Plastic Bending in the Reverse Direction

#### 2.2.6. Elastic Springback and the First Stable State

_{1.geom,y}is the curvature of the first stable state and m

_{I,y}is the moment along the Y-axis at the flattened shell after the second bending. The shell geometry and stress distribution at first stable state are shown in Figure 9.

#### 2.2.7. Second Stable State

_{2.geom,x}is the curvature of the first stable state and m

_{I,x}is the moment along the Y-axis at the flattened shell after the second bending. The shell geometry and stress distribution at second stable state are shown in Figure 10.

## 3. Results and Discussion

_{1}and R

_{2}was performed. The model study was performed at bending radii sequences of R

_{1}= {6, …, 12} mm and R

_{2}= {6, …, 12} mm with a step size of 0.1 mm. The semianalytical model was calculated using the software Python. The used material data of steel grade 1.1274 (AISI 1095) is taken from [9] and summarized together with the constants of the extended Voce equation [9] in Table 1.

_{0}is the initial hardening rate, θ

_{1}is the asymptotic hardening rate, and (σ

_{0}+ σ

_{1}) is the back-extrapolated yield stress, with σ

_{0}as the initial yield stress.

_{1}= {6, 7, 8, 10} mm and R

_{2}= {6, 7, 8, 10} mm and has already been partially published in [9]. In the following sections, different features of the shell will be discussed. Firstly, the bistability properties regardless of the realized curvature radii will be compared. Thereafter, the experimental and calculated curvature radii of shells at stable states will be studied. Finally, the calculated residual stress distribution for a bistable, fully closed shell, produced with bending radii combination R

_{1}= R

_{2}= 6 mm, will be compared with an experimentally measured distribution.

#### 3.1. Bistability of the Shell

_{1}= 7 mm, R

_{2}= 6 mm) and (R

_{1}= 8 mm, R

_{2}= 7 mm), experiments and semianalytical calculations do not coincide. Both combinations are situated almost at the boundary, separating bistable and monostable shells. Nevertheless, a more accurate yield stress determination could help to solve this issue.

#### 3.2. Radii of Stable Geometries

^{2}were incrementally bended. Opposite to experiments, in the semianalytical model, the size of the shell is considered as infinite and there is only one single bending around each axis. Despite this difference, a comparison of experiment and model regarding the final curvatures is done. In this way, the general tendencies concerning the influence of the bending radii on the resulting curvature could be compared. In Figure 12, the shell curvatures of first (a) and second (b) stable states depending on the second bending radius R

_{2}at a fixed first bending radius R

_{1}= 6 mm are depicted. It can be seen that the calculated results are in good agreement with the experimental values, despite the fact that calculated radii are higher than the experimental ones. It should also be mentioned that only for a bending radii combination of R

_{1}= R

_{2}= 6 mm, a fully closed bistable shell was produced. Due to the shell width of 300 mm, the maximal radius of a fully closed shell is equal to 47.75 mm, which is mentioned in Figure 12, Figure 13, Figure 14 and Figure 15 by a grey horizontal line. As the second bending radius increases, the curvature of the first stable state increases and the curvature of the second stable state decreases, both in experiment and in model.

_{1}= 7 mm. As was mentioned before, the main difference between calculated and experimental results is derived for the combination of (R

_{1}= 7 mm, R

_{2}= 6 mm). During experiments, the bistability was not achieved. For this part of the parameter study, it should be also noted that the calculated shell geometries have smaller curvature radii for almost all bending radii combinations. None of the experiments reached a fully closed curvature of bistable shell. In general, the curvature of the first stable state increases and the curvature of the second stable state decreases by increasing the second bending radius.

_{1}= 8 mm. The bistable behavior of the shell structure has been achieved during the experiments only for a bending radii of R

_{2}= 8 mm and R

_{2}= 10 mm. At the same time, the model predicts that the bistable properties will not be achieved for R

_{2}= 6 mm. However, the radii of curvature for both stable states are too high to achieve a fully closed shape of bistable shell.

_{1}= 10 mm. The bistable behavior of the shell structure has been achieved during the experiments and model calculation only for a bending radii combination of R

_{1}= R

_{2}= 10 mm. This could be seen also in Figure 15b, where only one curvature of second stable state for experimental results is marked. However, the radii of curvature for both stable states are too high to achieve a fully closed shape. For the bending radii combinations with R

_{2}< 10 mm, only small deviations between calculated and produced first stable states radii can be found. However, for the bending radii combination of R

_{1}= R

_{2}= 10 mm, a huge difference between the model and the experiment is obvious. The reason for this difference could be the proximity of the maximum plastic stress at this bending curvature to the initial yield stress of the material.

#### 3.3. Residual Stress Distribution

_{1}= R

_{2}= 6 mm. Stress values were measured from the surface until the middle of the shell with the step size of 0.02 mm for both stable geometries. The measured results are given in Figure 16 and show a good qualitative agreement between the model and the experimental results. Nevertheless, there are still deviations in the absolute values.

## 4. Conclusions

_{1}= R

_{2}= 6 mm. Moreover, a fully closed, metallic bistable shell was achieved within experiments with this bending radii combination.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | normal vector to the yield surface |

C | material stiffness matrix |

C_{t} | material tangent stiffness matrix |

E | Young’s modulus |

E_{t} | tangent modulus |

f | yield function |

H | hardening parameter |

I | identity matrix |

k | curvature of cylindrical surface |

k_{1}, k_{2} | first and second bending curvatures, respectively |

m | resultant bending moments per unit length |

n | number of increments |

R | bending radius |

t | shell thickness |

z | through-thickness coordinate measured from mid plane |

z_{cr} | critical depth |

ε | vector of principal strain components |

ν | Poisson’s ratio |

σ | vector of principal stress components |

σ_{i} | stress vector at step i |

σ_{v} | von Mises effective stress |

σ_{0} | yield stress in pure tension |

Subscripts | |

()_{el} | elastic |

()_{pl} | plastic |

()_{{A…J}} | at bending process point {A…J} |

()_{j.geom} | at j stable state |

Operators | |

()^{T} | transpose of the matrix |

()^{-1} | inverse of the matrix |

Δ | change between two consecutive steps |

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**Figure 2.**Scheme of the production process of bistable shells (

**a**) and schematic representation of experimental setup (

**b**) [9].

**Figure 3.**Bistable tube (produced by incremental die-bending with R

_{1}= R

_{2}= 6 mm) in deployed state as the first stable state (

**a**) and (partially) folded configuration as the second stable state (

**b**) [9].

**Figure 5.**Shell geometry (

**a**) and stress distribution along the shell thickness (

**b**) after first bending.

**Figure 8.**Shell geometry (

**a**) and stress distribution along the shell thickness (

**b**) after second bending.

**Figure 9.**Shell geometry (

**a**) and stress distribution along the shell thickness (

**b**) at first stable state.

**Figure 10.**Shell geometry (

**a**) and stress distribution along the shell thickness (

**b**) at second stable state.

**Figure 11.**Results regarding bistability for different combinations of bending radii obtained by experiments and semianalytical calculation.

**Figure 12.**Radii of curvature for first (

**a**) and second (

**b**) stable state after first bending operation with a radius R

_{1}= 6 mm and various second bending radii R

_{2}.

**Figure 13.**Radii of curvature for first (

**a**) and second (

**b**) stable state after first bending operation with a radius R

_{1}= 7 mm and various second bending radii R

_{2}.

**Figure 14.**Radii of curvature for first (

**a**) and second (

**b**) stable state after first bending operation with a radius R

_{1}= 8 mm and various second bending radii R

_{2}.

**Figure 15.**Radii of curvature for first (

**a**) and second (

**b**) stable state after first bending operation with a radius R

_{1}= 10 mm and various second bending radii R

_{2}.

**Figure 16.**Residual stresses over the shell cross section at first (

**a**) and second (

**b**) stable state (R

_{1}= R

_{2}= 6 mm).

**Table 1.**Averaged properties of steel 1.1274 obtained from quasi-static tensile tests and values of constants for the extended Voce equation [9].

Young’s Modulus, E | Initial Yield Stress, σ_{0} | σ_{1} | θ_{0} | θ_{1} |
---|---|---|---|---|

198,540 MPa | 1685 MPa | 128.9 MPa | 32,639.8 MPa | 4863.4 MPa |

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**MDPI and ACS Style**

Pavliuchenko, P.; Teller, M.; Grüber, M.; Hirt, G.
A Semianalytical Model for the Determination of Bistability and Curvature of Metallic Cylindrical Shells. *J. Manuf. Mater. Process.* **2019**, *3*, 22.
https://doi.org/10.3390/jmmp3010022

**AMA Style**

Pavliuchenko P, Teller M, Grüber M, Hirt G.
A Semianalytical Model for the Determination of Bistability and Curvature of Metallic Cylindrical Shells. *Journal of Manufacturing and Materials Processing*. 2019; 3(1):22.
https://doi.org/10.3390/jmmp3010022

**Chicago/Turabian Style**

Pavliuchenko, Pavlo, Marco Teller, Markus Grüber, and Gerhard Hirt.
2019. "A Semianalytical Model for the Determination of Bistability and Curvature of Metallic Cylindrical Shells" *Journal of Manufacturing and Materials Processing* 3, no. 1: 22.
https://doi.org/10.3390/jmmp3010022