# Modelling a Response of Complex-Phase Steel at High Strain Rates

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}, which can be experimentally determined on universal testing machines. However, to reach strain rates of an order of magnitude 1000 s

^{−1}or more, different experimental arrangements are needed. The most popular among them are a split Hopkinson bar test, a Taylor impact test, and a test of projectile shooting into a deformable body. While the parameters of the material models can be relatively simply determined from the tensile tests, this task becomes a challenge for the split Hopkinson bar test and the two impact tests. For this purpose, different reverse-engineering approaches can be followed [3,4]. Hernandez et al. [3] have proposed a dynamic characterisation computational procedure to determine the material parameters from a single Taylor impact test. The procedure involves the formulation and solution of an inverse problem to determine the Cowper–Symonds parameter of metals using the silhouette of a deformed Taylor specimen as input. Skrlec and Klemenc [4] have shown how the material parameters of the Johnson–Cook material model can be estimated from the projectile shooting test using the Taguchi design of experiments combined with explicit dynamic simulations, response surface for the cost function, and a numerical optimisation scheme.

^{−1}. Q390D had a significant rate of strain hardening. The yield and ultimate strength of the Q390D steel increased significantly with the increasing strain rate. The quasi-static strain rate was 0.00083 s

^{−1}. The strain-rate response was predicted by fitting the Cowper–Symonds model to the experimental data. Gupta et al. [6] studied the deformation behaviour and notch sensitivity of a super duplex stainless steel at different strain rates and temperatures. Qin et al. [7] investigated the mechanical behaviour of dual-phase high-strength steels (DP700 and DP500) under high strain rate tensile loading. The parameters of the Johnson–Cook material model were determined from experimental data well in the plastic zone. Digital image correlation was used with high-speed photography to study the strain localisation in the tensile specimens at high strain rates. Hu et al. [8] investigated the dynamic tensile characteristics of TRIP600, TRIP800, DP600, and DP800 steels at strain rates from 0.003 to 200 s

^{−1}. They obtained quantitative results for the increase in the flow stress as a function of the strain rate for both observed types of steel sheets. The results show that the DP-type steel sheets are more sensitive to the strain rate when compared to the TRIP-type steel sheets at the intermediate strain rates. They also concluded that the fracture elongation of the TRIP-type steel sheets decreases with increasing strain rates from 0.003 to 0.1 s

^{−1}but then increases up to the strain rate of 100 s

^{−1}due to the local strain rate hardening. The elongation of the DP-type steel sheets increases monotonically as the strain rate increases in contrast to the classical conjecture. Yu et al. [9] made quasi-static and dynamic tensile experiments for DP600 steel for the strain rate range between 10

^{−4}and 10

^{3}s

^{−1}by using conventional testing apparatus and the BTIA test. The results show that an obvious strain-rate-dependent mechanical behaviour exists for DP600 steel. The yield stress values at high strain rates are nearly twice those at low strain rates. The yield mechanisms are also different at high and low strain rates. The upper and lower yield points obviously exist at high strain rates, which is different from the material response at low strain rates. Shang et al. [10] studied the strain-rate and stress-state-dependent ductile fracture model of a S690 high-strength steel. They discovered that the strain rate effect of high-strength steel at the range of 10

^{−3}–10

^{3}s

^{−1}is well described by the Cowper–Symonds strain-rate-hardening criterion. They also modified the Johnson–Cook constitutive model using the Cowper–Symonds strain-rate-hardening description. Yang et al. [11] investigated the strain-rate-dependent behaviour of the S690 high-strength structural steel at intermediate strain rates. The tests indicated that the strain rate had an obvious influence on the behaviour of S690 steel, i.e., both the yield stress and tensile strength increased remarkably with increasing strain rate. The test results were then used to determine the coefficients of two commonly used strain-rate-dependent material models, the Cowper–Symonds model and the Johnson–Cook model. Wang et al. [12] researched the dynamic deformation and fracture mechanisms of Ti6Al4V over a wide strain rate range from quasi-static up to 10

^{4}s

^{−1}. The effects of the strain rate on the material deformation and associated fracture mechanism are discussed. The results demonstrate a strong strain-rate sensitivity of damage evolution of Ti6Al4V. Mahalle et al. [13] predicted static and dynamic flow stress behaviour of Inconel 718 alloy based on a modified Cowper–Symonds model, which was observed to provide better prediction in terms of statistical parameters for Inconel 718. Recently, studies with machine-learning-based modelling of strain rate and temperature effects were conducted [14,15].

## 2. Materials and Methods

#### 2.1. Tensile Stress–Strain Curves at Different Low-Strain Rates

^{−1}, 0.028 s

^{−1}, 0.14 s

^{−1}, 2.5 s

^{−1}, 5.2, and 5.4 s

^{−1}. At the lowest strain rate, strains were measured with an MTS 834.11F-24 mechanical extensometer, MTS Systems, Eden Prairie, MN, USA. At the same time, strains were also determined using the digital image correlation (DIC) method. For this purpose, the speckled colour pattern on the specimen was recorded with an i-SPEED 508 high-speed camera from IX Cameras, Rochford, UK. The sampling rate varied according to the loading rate and reached 4096 frames per second at the highest loading rate of 5.4 s

^{−1}. The camera’s frame rate was synchronised with the data acquisition sampling rate of the MTS testing machine. Digital image processing was carried out with Dantec Istra4D V4.10 software (https://www.dantecdynamics.com/new-istra4d-v4-10-software-release/, accessed on 30 April 2024). For the tests at higher strain rates, only the DIC method was used to estimate the strains.

^{−1}) were transformed using Equations (1) and (2) until necking. The parts of the curve after the necking onset were obtained with a connection between the point of ultimate tensile stress and the point, which was calculated using the measurement of the final specimen cross-section and force right before the fracture.

_{0}is an initial cross-section, and A and F are cross-section and measured force in recorded time frame, respectively.

^{−1}curve is presented together with its transformed true stress–true strain curve. By comparing the transformed true stress–true strain curve and the measured true stress–true strain curve, it can be concluded that the material characteristic continues with an almost linear shape after the necking occurs until the fracture of the specimen. The stress and strain at this point were calculated using the final specimen’s cross-section and loading force just before the fracture occurred, which coincided with the end of the direct true stress–true strain curve.

#### 2.2. Experimental Determination of Material Behaviour at High Strain Rates

#### 2.3. Identification of the Strain-Rate-Dependent Material Parameters

- A rough estimation of the three parameters SIGY, C, and p was first conducted with a grid-search method for the tensile tests at low strain rates.
- After the first phase, the SIGY parameter was fixed, and the two parameters, C and p, were fine-tuned using a reverse-engineering approach combined with a genetic algorithm optimisation procedure in the second phase.

#### 2.3.1. PHASE 1—Rough Estimation on the Basis of the Tensile Tests at a Low Strain Rate

_{p0.2}value; i.e., for each plastic flow curve, the knee point of the curve was shifted to zero. The average value of the shifted curves represented the curve ${\sigma}_{Y}\left({\epsilon}_{pl}\right)$ in Equation (5).

^{−10}s

^{−1}in our case. Then, the corresponding flow curve ${\sigma}_{Y,S}\left({\epsilon}_{pl}\right)$ needs to be defined. We decided to model it with a piecewise linear function, which started below the knee point of the measured true stress–true strain curves—see Figure 9. This means that the SIGY parameter becomes only a parameter that needs to be estimated and is not directly related to the material’s R

_{p}

_{0.2}value. Moreover, we have discovered that Equation (5) models the material response very poorly if the SIGY parameter is set to the value of R

_{p}

_{0.2}for the smallest strain rate measured. Since there was no clear point of deviation from linearity in the case of the high-strain rate experiments, an inverse analysis was followed in the second phase in order to obtain the final estimates of the three parameters from Equation (5).

_{LSR}:

_{LSR}was obtained for the following combination of the Cowper-Symonds parameters: SIGY = 400 MPa, C = 50 and p = 25. At this point, the value SIGY = 400 MPa was fixed, and the estimations of the two parameters, C and p, were refined. Again, the grid search algorithm was used to minimise the combined cost function from Equation (6), but now, it has a different division of the two parameters; see Table 4.

_{LSR}cost function from Equation (6) was calculated for each combination of the parameters C and p in Table 3. The SIGY parameter was kept constant at 400 MPa. The surface of the summarised cost function SSQD

_{LSR}is presented in Figure 10. Now, the most optimal estimates of the two parameters C and p were: C = 210 and p = 30.

#### 2.3.2. PHASE 2—Enhanced Estimation Based on the Reverse-Engineering Approach

_{LSR}from Equation (6) was applied. It was calculated for each combination of the material parameters from Table 4. For the high-strain-rate experiments, the cost function was calculated as follows:

_{ij}

_{,sim}and the coordinate of the maximum indentation Y

_{ij,sim}from the simulated results for each boundary/initial condition j = 1, …, 9 and each combination of the material parameters i = 1, …, n; n = 49.

_{HSR,i}for the high strain-rate experiments for each combination of the material parameters from Table 4:

_{i}for low- and high-strain-rate experiments:

_{LSR}from Equation (6) was divided by one million to obtain the same order of magnitude as the high-strain-rate cost function SSQD

_{HSR}from Equation (7). A sensitivity analysis for the weight u was carried out, in which this weight was varied between 0.2 and 0.8. It turned out that the results of the optimisation process were the same if this weight was between 0.3 and 0.7. For this reason, the final value of 0.5 was selected for the weight u.

_{i}. The adopted form of the cost function was taken from the work of Škrlec and Klemenc [4]. The global trend of the cost function CF

_{i}was first approximated with a polynomial of the third order:

_{i}was calculated for each combination of the material parameters C and p:

_{i}were finally interpolated using the sum of the weighted two-dimensional Gaussian functions with diagonal covariance matrices

**S**

_{i}:

**S**

_{i}are equal to the squared values of the half-distances to the nearest neighbour in (C, p) space along the C or p axis. The weighting parameters b

_{i}in Equation (12) were calculated with the following matrix equation:

## 3. Results

^{−1}for the highest ball velocity. These strain rates could not be measured directly but were determined from the results of numerical simulations for the most optimal values of the Cowper–Symonds parameters.

^{−1}and p = 30.4. In Figure 14, an agreement between the selected modelled and measured yield curves from Section 2.1 is presented. For the modelled curves, the value of SIGY = 400 MPa was considered. It can be concluded that the modelled and experimentally determined yield curves are in very good agreement. The agreement between the results of numerical simulations and measured deformations of the plate after the ball impact is good. A relative error between 0.8% and 8.4% was obtained for the Y-coordinate of the maximum indentation depth, and a relative error between 0.7% and 22.5% was obtained for the maximum indentation depth.

## 4. Discussion

^{7}s

^{−1}can be found in the literature if there are high strain rates in the interval above 10

^{3}s

^{−1}. In contrast, the values of parameter C are six orders of magnitude lower if there are strain rates only up to 0.1 s

^{−1}. Moreover, the two parameters, p and C, are strongly correlated. This is a consequence of the shape of the cost function. We have discovered before [4] that many different pairs of values p and C result in similar material responses, especially in the high-strain-rate domain. For this reason, it is very important to estimate the strain-rate-dependent material parameters over a wide range of strain rates. Finally, the estimated values of the parameters p and C also depend on the formulation of the Cowper–Symonds model. In our case, the visco-plastic formulation was used in Equation (5) to allow for parallel transformation of the yield curve along the stress axis. If another formulation of the Cowper–Symonds model was used (i.e., with scaling of the whole yield curve as a function of the strain rate; see [1,2]), different values of the parameters p and C would be obtained. At this point, we should also mention that the optimal values of the parameters p and C depend on the selected value of the SIGY parameter. However, to model the material response as well as possible, it is important that the complete flow curve from Equation (5) generalises the material response at low and high strain rates. Even though this means that the parameter SIGY does not reflect the R

_{p0.2}value from the experiments.

## 5. Conclusions

^{−1}was obtained. The true stress–true strain curves were obtained based on transformation equations and directly measured using high-speed camera recordings with full HD resolution and the digital image correlation method. Both procedures gave very similar curve shapes for the whole range of strains up to fracture.

- The estimated values of the Cowper–Symonds parameters p and C differ somewhat from the data in the literature for similar materials.
- Parameter estimates are strongly dependent on the applied testing methods and strain rates.
- Due to the shape of the multi-criteria cost function, a wide domain of the combinations for the material parameters p and C yield acceptable results.
- Estimated values of the material parameters also depend on the mathematical formulation for the strain-rate effects.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Comparison of engineering stress–engineering strain curve and the transformed true stress–true strain curve.

**Figure 8.**The 3D digital image correlation (DIC) system with three cameras (

**a**), three images of the deformed specimens (

**b**), the definition of the coordinate system (red, green and blue points) (

**c**), marking of points (blue) in the regions of interest (

**d**).

**Figure 9.**Used flow curve ${\sigma}_{Y,S}\left({\epsilon}_{pl}\right)$ at quasi-static strain rates.

**Table 1.**Chemical composition of SZBS800 steel [16].

C | Si | Mn | P | S | Al | B | Cu |
---|---|---|---|---|---|---|---|

max. | max. | max. | max. | max. | min. | max. | max. |

0.18% | 1.00% | 2.20% | 0.05% | 0.01% | 0.015–1.2% | 0.005% | 0.2% |

Specimen Number | Impact Angle a _{j} [°] | Ball Velocity v _{j} [m/s] |
---|---|---|

20210503_H800-1 | 0 | 111.5 |

20231108_H800-1 | 0 | 133.3 |

20210504_H800-1 | 0 | 155.3 |

20231108_H800-3 | 20 | 111.2 |

20231108_H800-2 | 20 | 132.6 |

20210714_H800-1 | 20 | 154.2 |

20210507_H800-2 | 35 | 113.1 |

20231110_H800-1 | 35 | 131.9 |

20210713_H800-1 | 35 | 153.5 |

SIGY [MPa] | C [s^{−1}] | p [/] |
---|---|---|

200 | 1 | 0.1 |

300 | 2.5 | 0.25 |

400 | 5 | 0.5 |

500 | 10 | 1 |

600 | 50 | 2.5 |

100 | 5 | |

250 | 10 | |

500 | 15 | |

2500 | 25 | |

10,000 | 50 | |

100 |

p [/] | C [s^{−1}] | p [/] | C [s^{−1}] |
---|---|---|---|

10 | 10 | 80 | 190 |

15 | 30 | 90 | 210 |

20 | 50 | 100 | 230 |

25 | 70 | 250 | |

30 | 90 | 270 | |

40 | 110 | 300 | |

50 | 130 | 600 | |

60 | 150 | 1000 | |

70 | 170 | 1500 |

p [/] | C [s^{−1}] | p [/] | C [s^{−1}] |
---|---|---|---|

15 | 70 | 35 | 390 |

20 | 150 | 40 | 470 |

25 | 230 | 45 | 550 |

30 | 310 |

Specimen Number | Impact Angle α _{j} [°] | Ball Velocity v _{j} [m/s] | Y-Coordinate of max. Indentation Y _{j,}_{exp} [mm] | Maximum Indentation Depth H _{j,}_{exp} [mm] |
---|---|---|---|---|

20210503_H800-1 | 0 | 111.5 | 29.762 | 5.768 |

20231108_H800-1 | 0 | 133.3 | 29.282 | 8.057 |

20210504_H800-1 | 0 | 155.3 | 29.716 | 8.497 |

20231108_H800-3 | 20 | 111.2 | 30.306 | 5.975 |

20231108_H800-2 | 20 | 132.6 | 29.575 | 6.940 |

20210714_H800-1 | 20 | 154.2 | 32.632 | 8.025 |

20210507_H800-2 | 35 | 113.1 | 32.325 | 5.726 |

20231110_H800-1 | 35 | 131.9 | 32.331 | 6.055 |

20210713_H800-1 | 35 | 153.5 | 33.821 | 6.911 |

Material Name | EN Code | Testing Method | p [-] | C [s^{−1}] |
---|---|---|---|---|

S690QL ultra-high-strength steel—EN 10025 [22] | 1.8931 | Low strain rates 10^{−3} s^{−1}: Zwick/Roell—Z50 (ZwickRoell GmbH & Co., Ulm, Germany)Medium strain rates 3 s ^{−1}–30 s^{−1}: Hydro-pneumatic machine;High strain rates 250 s ^{−1}–950 s^{−1}: Split Hopkinson Tensile Bar | 2.3 to 4.49 | 121,783 to 212,352 |

Hot-forming steel 22MnB5 [23] | 1.5528 | Low strain rates 10^{−3} s^{−1}: WDW-100E uniaxial tensile testing machine;High strain rates 2000 s ^{−1}–4000 s^{−1}: Split Hopkinson Pressure Bar | 3.2050 | 6277.8 |

S690 high-strength structural steel [11] | 1.3964 | Low strain rates 10^{−3} s^{−1}: universal electromechanical testing machine;Intermediate strain rates 10 s ^{−1}–200 s^{−1}: High-speed tensile testing machine | 6.1–6.7 | 1.2 × 10^{7} to 3.3 × 10^{7} |

high-strength reinforcing steel HTRB600E [24] | Low strain rates: below 2 × 10^{−3} s^{−1}: electromechanical universal testing machine;Elevated strain rates 0.018–585 s ^{−1}: high-speed servo-hydraulic tensile testing machine Zwick/Roell HTM5020 | 5.925 to 6.027 | 1.83 × 10^{7} to 8.67 × 10^{7} | |

Q390D steel [5] | Low strain rates 8.4 × 10^{−4} s^{−1}: INSTRON 5569 uniaxial tensile testing machine (Instron, Norwood, MA, USA);High strain rates 837.1 s ^{−1}–3368.1 s^{−1}: Split Hopkinson pressure Bar | 1.31 to 2.4 | 3097.14 to 3861.44 | |

Duplex Stainless Steel 2507 [6] | 1.4410 | Low strain rates 0.0001–0.1: electromechanical universal testing machine (Zwick/Roell Z250) | 1.7566 | 2.68274 |

MP800HY Steel [25] | Low strain rates 10^{−3} s^{−1}: Zwick/Roell—Z50;Medium strain rates 5 s ^{−1}–25 s^{−1}: Hydro-pneumatic machine;High strain rates 250 s ^{−1}–750 s^{−1}: modified Hopkinson Bar | 2.2146 | 32,338 | |

steel B500A [26] | Low strain rates 10^{−4} s^{−1}: universal electromechanical testing machine;High strain rates 250 s ^{−1}–1000 s^{−1}: Split Hopkinson Tensile Bar | 1.654 to 4.4677 | 5.574 to 525.444 | |

Steel HRB500E [27] | Low strain rates 2.5 × 10^{−4}–5.3 × 10^{−1} s^{−1}: universal electromechanical testing machine;High strain rates 0.1 s ^{−1}–550 s^{−1}: Zwick/Roell HTM5020 servo-hydraulic high-speed testing machine | 4.906 | 264,713 | |

SZBS800 steel | 1.0998 | Low strain rates: 100 kN MTS Landmark testing machine; High strain rates: shooting ball into flat specimen | 30.4 | 172.4 |

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## Share and Cite

**MDPI and ACS Style**

Škrlec, A.; Kocjan, T.; Nagode, M.; Klemenc, J.
Modelling a Response of Complex-Phase Steel at High Strain Rates. *Materials* **2024**, *17*, 2302.
https://doi.org/10.3390/ma17102302

**AMA Style**

Škrlec A, Kocjan T, Nagode M, Klemenc J.
Modelling a Response of Complex-Phase Steel at High Strain Rates. *Materials*. 2024; 17(10):2302.
https://doi.org/10.3390/ma17102302

**Chicago/Turabian Style**

Škrlec, Andrej, Tadej Kocjan, Marko Nagode, and Jernej Klemenc.
2024. "Modelling a Response of Complex-Phase Steel at High Strain Rates" *Materials* 17, no. 10: 2302.
https://doi.org/10.3390/ma17102302