# Upgrade of Biaxial Mechatronic Testing Machine for Cruciform Specimens and Verification by FEM Analysis

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- -
- Slow speed for the experiment;
- -
- Fast mode for piston position adjustment.

## 2. Materials and Methods

#### 2.1. Technical Realization of the Hydraulic Loading Device

- A hydraulic device for biaxial loading of the cruciform specimens;
- A dynamic strain gauge measurement device for sensing the evolution of the loading forces in the cruciform specimen arms using dynamometers with resistive strain gages;
- An extensometer for measuring the deformation in the central part of the cruciform specimen under biaxial tensile loading.
- The hydraulic loading device consists of the following main parts:
- A lamellar hydrogenerator SA33-100;
- Hydraulic cylinders;
- Clamping jaws;
- A guide body;
- Throttling valves;
- A main valve;
- Distributors of working fluid pressure;
- Electrical switches.

#### 2.1.1. Lamellar Hydrogenerator SA3-100

- Power W = 7.5 kW;
- Maximum working fluid flow Q = 40 dm
^{3}·min^{−1}; - Working temperature 50 °C;
- Maximum working pressure of 16 MPa.

#### 2.1.2. Hydraulic Cylinders and Hoses

- Direction X: D = 63 mm, d = 32 mm, t = 6.5 mm, and lift= 250 mm;
- Direction Y: D = 60 mm, d = 32 mm, t = 6.5 mm, and lift= 100 mm.

#### 2.1.3. Clamping Jaws and Guide Body

#### 2.1.4. Parameters of the Main and Throttle Valves

- Maximum working fluid flow: Q = 30 dm
^{3}·min^{−1}; - Maximum working pressure: P= 35 MPa;
- Maximum working temperature: T = 80 °C.

**Figure 3.**The hydraulic scheme of the loading device, where: 1-Manually operated Needle Valve, 2-Manually Operated Check Valve, 3-Manually Operated Check Valve, 4-Solenoid Operated Modular Valve,4-Way, 2-Position Crossover Offset, 5-Solenoid Operated Modular Valve, 4-Way, 3-Position Tandem Center, 6-Solenoid Operated Modular Valve,4-Way, 3-Position Tandem Center.

- Maximum working fluid flow: Q = 1.6 dm
^{3}·min^{−1}; - Maximum working pressure: P= 32 MPa.

#### 2.1.5. Electrical Switches

#### 2.2. Load Mechanism Improvement through Regulator Design

- Application of force to the testing specimen;
- Ratio regulation of deformations and mechanical stress;
- That the time change in the deformation will not be greater than prescribed in the relevant norm.

#### 2.2.1. Regulator Design in a Block Diagram

#### 2.2.2. Modernization of the Hydraulic Control Circuit

## 3. Mathematical Model

#### 3.1. Mathematical Model of the Electrohydraulic Actuator

#### 3.2. Mathematical Model of the Proportional Pressure-Reducing Valve

_{i}is the sensitivity and k

_{p}is the compliance of the valve, Equation (8).

#### 3.3. Mathematical Model of the Cruciform Specimen

#### 3.4. Design of the Resulting Block Diagram of the Control Circuit

_{m}, to be in the range of 30° to 60°. In our case, we choose the value in Equation (14).

## 4. Results

#### 4.1. Design of the Resulting Block Diagram of the Control Circuit

#### 4.2. Nonlinear FEM Simulation of the Specimen Based on the Plasticity Anisotropic Material Model

- Several mesh configurations were checked, and the dependence of the change in the peak stress on the change in degrees of freedom was realized, together with the dependence of the peak stress on the change in the number of elements. In this way, the convergence of the mesh was verified, Figure 25;
- The upper limit of the maximum stress was set at which the calculation was still stable without the need for drastic changes in the convergence tolerance and singularity elimination factor;
- The minimum element size was determined based on geometric parameters at 0.078 mm, and the maximum element size was determined based on convergence properties at 2.62 mm.

#### 4.3. Experimental Verification of the Closed-Loop Control System and the Plasticity Properties of the Cruciform Specimen

_{(}

_{20)}and r

_{(5)}give the values of the coefficients of the normal anisotropy for 20% and 5% deformations, respectively. Rp

_{0.2}is the yield stress (the stress at which the relative specimen elongation reaches 0.2%), A

_{80}is the tensibility, and R

_{m}is strength limit. Figure 37 shows typical actual tensibility diagrams of KOHAL E 210 IZ. As can be seen from the figures, the quality of KOHAL E 210 IZ in this region of plastic deformation is essentially free of stiffening. In the experiment, the arms of the cruciform specimens were retracted at constant ratios of 1:5, 1:2, 1:1, 2:1, and 5:1 piston speeds of the hydraulic loading device. Figure 38a shows the typical dependencies ${\epsilon}_{2}={\epsilon}_{2}\left({\epsilon}_{1}\right)$ for the selected speed ratios, and Figure 38b shows the typical dependencies ${\sigma}_{2}={\sigma}_{2}\left({\sigma}_{1}\right)$.

## 5. Conclusions

- -
- Design of the structure of the control closed-loop circuit at the functional and object level;
- -
- The detailed design of the electrohydraulic part of the loading device at the object level;
- -
- Design of mathematical models within individual elements of the control closed-loop circuit;
- -
- The final closed-loop block diagram and verification of the controlled system;
- -
- Design of a regulator of the control circuit in the frequency domain using the method of shaping the frequency response of an open-loop control circuit;
- -
- The computer simulation of a control closed-loop circuit in the MATLAB/Simulink environment and its evaluation;
- -
- FEM simulation of the cruciform specimen;
- -
- Experimental verification of the closed-loop control system and plasticity properties of the cruciform specimen.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Šimčák, F.; Trebuňa, F.; Hanušovský, J. Evaluation of Plastic Properties of Sheets in Plane Stress States; Skalský dvůr: Lísek, Czech Republic, 2005; ISBN 80-214-2941-0. [Google Scholar]
- Šimčák, F. Inovačné trendy pri zvyšovaní únosnosti karosérií automobilov. Acta Mech. Slovaca
**2003**, 1, 13–24. [Google Scholar] - Gambin, W. Plasticity and Textures; Kluver Academic Publisher: Dordrecht, The Netherlands, 2001. [Google Scholar]
- Mises, R.V. Mechanics of plastic deformation of crystals 592. Zeitsch.Angew. J. Appl. Math. Mech.
**1928**, 8, 161–185. [Google Scholar] - Hill, R. Mathematical Theory of Plasticity; Oxford University Press: Oxford, UK, 1956. [Google Scholar]
- Hill, R. Constitutive modelling of orthotropic plasticity in sheet metals. J. Mech. Phys. Solids
**1990**, 38, 405–417. [Google Scholar] [CrossRef] - Hill, R.A. User-friendly theory of orthotropic plasticity in sheet metals. Int. J. Mech. Sci.
**1993**, 15, 19–25. [Google Scholar] [CrossRef] - Hosford, W.F. A 29 eneralized isotropic yield criterion. J. Appl. Mech.
**1972**, 39, 607–609. [Google Scholar] [CrossRef] - Gotoh, M. A theory of plastic anisotropy based on a yield function of fourth order. Int. J. Mech. Sci.
**1977**, 19, 505–520. [Google Scholar] [CrossRef] - Barlat, F.; Lege, D.J.; Brem, J.C. A six-component yield function for anisotropic materials. Int. J. Plast.
**1991**, 7, 693–712. [Google Scholar] [CrossRef] - Pöhlandt, K.; Banabic, D.; Balan, T.; Comsa, D.S.; Müller, W. A new criterion for anisotropic sheet metals. In Proceedings of the 8th International Conference Achievements in the Mechanical and Materials Engineering, Gliwice, Poland; 1999; pp. 33–36. [Google Scholar]
- Šimčák, F.; Hanušovský, J.; Berinštet, V. Experimental determination of yield locus of steel sheets by biaxial tensile test. Acta Mech. Slovaca
**2006**, 1, 535–542. [Google Scholar] - Boehler, J.P.; Demmerle, S.; Koss, S. New direct biaxial testing machine for anisotropic materials. Exp. Mech.
**1994**, 34, 1–9. [Google Scholar] [CrossRef] - Makinde, A.; Thibodeau, L.; Neale, K.W. Development of an Apparatus for Biaxial Testing Using Cruciform Specimens. Exp. Mech.
**1992**, 32, 138–144. [Google Scholar] [CrossRef] - Sheet Metal Structures. Maintenance Manual (MM) or Structural Repair Manual. Available online: https://membefiles.freewebs.com/76/87/106238776/documents/2%20SHEET%20METAL%20STRUCTURES.pdf (accessed on 6 July 2021).
- Kuwabara, T.; Ikeda, S.; Kuroda, K. Measurement and Analysis of Work Hardening of Sheet Metals Under Plane—Strain Tension. J. Mater. Process. Technol.
**1998**, 80–81, 97–102. [Google Scholar] - Shimamoto, A.; Shimomura, T.; Nam, J.H. The Development of a Servo Dynamic Biaxial Loading Device. Key Eng. Mater.
**2003**, 243–244, 99–104. [Google Scholar] [CrossRef] - Pereira, A.B.; Fernandes, F.A.O.; de Morais, A.B.; Maio, J. Biaxial Testing Machine: Development and Evaluation. Machines
**2020**, 8, 40. [Google Scholar] [CrossRef] - Chen, C.; Li, Z.; Xu, C.; Zhu, Z.; Zou, S. Variation of Fracture Toughness with Biaxial Load and T-Stress under Mode I Condition. Appl. Sci.
**2022**, 12, 9319. [Google Scholar] [CrossRef] - Chen, J.; Zhang, J.; Zhao, H. Quantifying Alignment Deviations for the In-Plane Biaxial Test System via a Shape-Optimised Cruciform Specimen. Materials
**2022**, 15, 4949. [Google Scholar] [CrossRef] [PubMed] - Ru, M.; Lei, X.; Liu, X.; Wei, Y. An Equal-Biaxial Test Device for Large Deformation in Cruciform Specimens. Exp. Mech.
**2022**, 62, 677–683. [Google Scholar] [CrossRef] - Wang, S.; Hou, C.; Wang, B.; Wu, G.; Fan, X.; Xue, H. Mechanical responses of L450 steel under biaxial loading in the presence of the stress discontinuity. Int. J. Press. Vessel. Pip.
**2022**, 198, 104662. [Google Scholar] [CrossRef] - Corti, A.; Shameen, T.; Sharma, S.; De Paolis, A.; Cardoso, L. Biaxial testing system for characterization of mechanical and rupture properties of small samples. HardwareX
**2022**, 12, e00333. [Google Scholar] [CrossRef] [PubMed] - Berinštet, V.; Tomčík, J. Modernization of the experimental workplace for evaluation of elastic-plastic properties of sheet metals in plane stress. Acta Mechanica Slovaca. Techonl. Univerzita-Stroj. Fak.
**2005**, 12, 69–76. [Google Scholar] - Medvecká-Beňová, S. Strength analysis of the frame of a trailer. Sci. J. Sil. Univ. Technol. Ser. Transport.
**2017**, 96, 105–113. [Google Scholar] [CrossRef] - Airmotive Specialties, Inc. Salinas, CA 93901. Available online: http://www.airmotives.com/index.html (accessed on 18 August 2021).

**Figure 7.**Diagram of the control circuit at the object level (1—metal cruciform sample (specimen), 2—hydraulic cylinders with hydraulic fluid lines for the x-axis, 3—hydraulic cylinders with hydraulic fluid lines for the y-axis, 4—proportional pressure-reducing valve PDR08P-01 for the x-axis, 5—proportional pressure-reducing valve PDR08P-01 for the y-axis, 6—strain gauge sensors, 7—measuring card RB-01-TU-002-100-E, 8—multifunctional measuring card MF 624 (MF 604) for data collection and computer processing).

**Figure 8.**Hydraulic scheme of a modernized hydraulic circuit of the loading equipment where, port 1 is the main oil flow from the pump, port 2 is the main oil flow to the valve, and port 3 is the oil outflow from the valve.

**Figure 11.**Block diagram of the hydraulic part of the hydraulic motor. Red lines are Input/Output (I/O) signals in this part of the block diagram.

**Figure 12.**Continuation of the block diagram of the hydraulic part of the hydraulic motor based on the modified Equation (6), with the red lines being I/O signals in this part of the block diagram.

**Figure 13.**Block diagram of the proportional valve, with red lines being I/O signals in this part of the block diagram.

**Figure 26.**Typical stress distributions for the force ratio Fx:Fy = 1 using the von Mises condition of plasticity in the final step of the simulation. Purple arrows show the direction of the load; green arrows show the virtual action of the slider.

**Figure 27.**Applying the iso clipping function to isolate values of stress which are more than the yield strength in the plastic region. Purple arrows show the direction of the load; green arrows show the virtual action of the slider.

**Figure 28.**On the left, stress distributions SX on the x-axis are equal to ${\sigma}_{x}$, and on the right, stress distributions SY on the y-axis are equal to ${\sigma}_{y}$ using von Mises condition. Purple arrows show the direction of the load; green arrows show the virtual action of the slider.

**Figure 29.**Central region for stress distribution analysis using the von Mises condition of plasticity in the final step of the simulation.

**Figure 30.**Typical strain distributions for the force ratio Fx:Fy = 1 in the final step of the simulation. Purple arrows show the direction of the load; green arrows show the virtual action of the slider.

**Figure 31.**On the left, strain distributions EPSX on the x-axis are equal to ${\epsilon}_{x}$, and on the right, strain distributions EPSY on the y-axis are equal to ${\epsilon}_{y}$ for the force ratio Fx:Fy = 1 in the final step of the simulation. Purple arrows show the direction of the load; green arrows show the virtual action of the slider.

**Figure 32.**Graph of the change in the stress in the node number 34331, reflecting the center of the cruciform specimen during the time in the simulation.

**Figure 33.**Graph of the change in the displacement in the node number 34331, reflecting the center of the cruciform specimen during the time in the simulation.

**Figure 34.**Figure 34. Graph of the characteristic which is based on ${\sigma}_{y}={\sigma}_{y}\left({\sigma}_{x}\right)$ stress, where SY is equal to ${\sigma}_{y}$, and SX is equal to ${\sigma}_{x}$.

**Figure 35.**Graph of stress changing with respect to displacement in the center of the specimen along the simulation’s steps.

**Figure 36.**On left side—comparison of the setpoint (green line) and the controlled variable (blue line). On right side—real output of the extensometer.

**Figure 38.**Addictions (

**a**) ${\epsilon}_{2}={\epsilon}_{2}\left({\epsilon}_{1}\right)$ (

**b**) ${\sigma}_{2}={\sigma}_{2}\left({\sigma}_{1}\right)$.

**Figure 39.**(

**a**) Experimentally determined points of plasticity curves of KOHAL E 210 IZ in stress ${\sigma}_{1}$ and ${\sigma}_{2},$ on (

**b**) experimentally determined points of plasticity curves shown in stress ${\sigma}_{1}/{\sigma}_{k90}$ and ${\sigma}_{2}/{\sigma}_{k90}$.

Selected Step of Simulation | Computed Value—Strain in Plastic Deformation (Center of Specimen) |
---|---|

In 100 | ${\epsilon}_{x}^{s}=0.001945$$,{\epsilon}_{y}^{s}=0.001944$ |

In 150 | ${\epsilon}_{x}^{s}=0.008585$$,{\epsilon}_{y}^{s}=0.008582$ |

In 200 | ${\epsilon}_{x}^{s}=0.01153$$,{\epsilon}_{y}^{s}=0.01152$ |

In 250 | ${\epsilon}_{x}^{s}=0.01615$$,{\epsilon}_{y}^{s}=0.01614$ |

In 463 (final) | ${\epsilon}_{x}^{s}=0.03418$$,{\epsilon}_{y}^{s}=0.03415$ |

Material | Thickness [mm] | Direction | R_{p0.2} [MPa] | R_{m} [MPa] | A_{80} [%] | r_{(20)} | r_{(5)} |
---|---|---|---|---|---|---|---|

KOHAL E 210 IZ | 1.00 | 0° | 239 | 353 | 37 | 0.91 | 0.86 |

45° | 254 | 352 | 38 | 1.03 | 1.09 | ||

90° | 267 | 358 | 38 | 1.09 | 1.17 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Miková, Ľ.; Prada, E.; Kelemen, M.; Krys, V.; Mykhailyshyn, R.; Sinčák, P.J.; Merva, T.; Leštach, L.
Upgrade of Biaxial Mechatronic Testing Machine for Cruciform Specimens and Verification by FEM Analysis. *Machines* **2022**, *10*, 916.
https://doi.org/10.3390/machines10100916

**AMA Style**

Miková Ľ, Prada E, Kelemen M, Krys V, Mykhailyshyn R, Sinčák PJ, Merva T, Leštach L.
Upgrade of Biaxial Mechatronic Testing Machine for Cruciform Specimens and Verification by FEM Analysis. *Machines*. 2022; 10(10):916.
https://doi.org/10.3390/machines10100916

**Chicago/Turabian Style**

Miková, Ľubica, Erik Prada, Michal Kelemen, Václav Krys, Roman Mykhailyshyn, Peter Ján Sinčák, Tomáš Merva, and Lukáš Leštach.
2022. "Upgrade of Biaxial Mechatronic Testing Machine for Cruciform Specimens and Verification by FEM Analysis" *Machines* 10, no. 10: 916.
https://doi.org/10.3390/machines10100916