# Estimating the Cowper–Symonds Parameters for High-Strength Steel Using DIC Combined with Integral Measures of Deviation

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}and impact ball tests with varying impact velocities that resulted in strain rates up to 450 s

^{−1}(The SZBS800 steel was used in our research because there was no high-strain rate data for this kind of steel in the literature). The impact ball test was a derivation of the impact test that was originally developed for polymer materials [16]. In this approach, the C and p parameters were estimated by minimising a combined cost function that was a superposition of (i) a sum-of-squares deviation between the simulated and measured σ-ε curves emanating from the tensile tests at low-strain rates and (ii) a deviation between the simulated and measured imprint depth and position from impact ball tests at high strain rates. The obtained result for the C and p parameters was compared to the available literature data for comparable high-strength steels:

- Multi-phase high yield strength steel (MP800HY), according to [28],

^{−1}[22] and 4000 s

^{−1}[25]. By comparing the methodologies and data from the literature, the following possible causes for such differences are listed below:

- The selected testing method and the achieved strain rates have a significant influence on the estimated parameters. In general, the higher the strain rates achieved, the higher the parameter C.
- The two parameters C and p are not independent but are strongly correlated.
- The considered strain-rate range influences the estimated parameters C and p.
- The forms of the applied Cowper–Symonds model may be different.
- How the cost function for estimating the strain-rate dependent parameters is defined is important.

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{−1}, a different experimental arrangement was used. This experiment was actually a modified version of a testing arrangement according to the ASTM D5420 standard [16]. A steel ball with a diameter of 13.6 mm and a weight of 10.6 g was shot at high velocity into a rectangular 1 mm thick plate-like specimen with a dimension of 98 × 60 mm

^{2}. The specimen was clamped along its two short edges in such a way that it had a free surface of 60 × 60 mm

^{2}. The ball used in the experiment was taken from a Schaeffler ball bearing (FAG before) [44]. According to their catalogue, steel balls are made from through-hardened rolling bearing steel 100Cr6 in accordance with [45], with tensile yield strength over 1700 MPa, tensile ultimate strength up to 2300 MPa, and 61 HRC of hardness. The ball was shot to the specimen with a pneumatic gun manufactured for this purpose. The gun outlet and the specimen were positioned in a safety chamber. For the impact tests, two parameters were varied: (i) the impact velocity (between 111 m/s and 155 m/s) and (ii) the impact angle (between 0° and 35°). Each impact experiment was recorded with the high-speed camera. Its recordings were used to improve the accuracy of the measured velocities from the gun’s photosensors. Altogether, nine impact tests were performed. For details, see Škrlec et al. [15]. In contrast to the low strain-rate tensile tests, it was difficult to determine the strain rate during the impact tests. With the help of the numerical simulations, which are explained later, it was estimated that the strain rate during the impact tests was between 300 s

^{−1}and 450 s

^{−1}.

#### 2.2. Modelling the Strain-Rate-Dependent Material Response

^{−10}s

^{−1}. The flow curve ${\sigma}_{Y,S}\left({\epsilon}_{pl}\right)$ in the MAT_24 material model was modelled with a piece-wise linear function starting below the knee—point-of the measured true-stress–true strain curve. Consequently, as can also be seen in Figure 4, it can be concluded that the SIGY parameter has nothing to do with real yield stress but is merely a material parameter that is applied to adjust a vertical shift of the flow curve as a function of the strain rate. Moreover, we discovered in our previous research [15] that Equation (3) models the material response very poorly if the SIGY parameter is set to the value of the yield stress for the smallest measured strain rate.

#### 2.3. Estimating the Material Parameters from the Experimental Data

- Phase 1: a rough estimation of the three parameters that is based on the low strain-rate tensile tests.
- Phase 2: fine-tuning of the C and p parameters using multi-criterion cost function combined with numerical optimisation and reverse engineering. In this phase, low-strain rate and high-strain rate experiments are considered.

_{LSR,i}for each combination k of the three parameters SIGY, C, and p:

^{−1}and p = 30. Further details on Phase 1 can be found in Škrlec et al. [15].

_{x}× n

_{x}matrix, which represents the reference image. The n

_{x}and n

_{y}are the number of rows and columns, respectively. Let $i\in 1,\dots ,{n}_{x}$ and $j\in 1,\dots ,{n}_{y}$ be the matrix entry indices. Each matrix entry i, j contains the value of displacement ${y}_{i,j}$ at the corresponding coordinate values x

_{i}and z

_{i}. The coordinates x

_{i}and z

_{i}are calculated as follows:

_{min}and z

_{min}give the coordinate origin values of respective coordinates and are dependent on coordinate placement while performing a simulation. The respective coordinate increment d

_{x}and d

_{z}can be calculated as:

_{max}and z

_{max}represent the specimen width and length, respectively. Hence, by increasing or lowering the number of rows n

_{x}and columns n

_{z}in a matrix, a more or less smooth image of displacement field is achieved, as depicted in Figure 8.

_{LSR,k}and ISE

_{k,l}should be combined together to form a multi-criterion cost function CF

_{i}. For this purpose, an average ${\mu}_{l}$ and a standard deviation ${\sigma}_{l}$ of the ISE

_{k,l}measure was first calculated for each high-strain rate test case l = 1,…, 9:

_{k,l}measures were standardised and summed together over the nine high-strain-rate test cases to form the SSQD

_{HSR,k}cost function for the high strain-rate experiments:

_{LSR,k}was evaluated anew for the $nu{m}_{comb}=119$ combinations of the C and p parameters in Table 3 and the 13 low-strain-rate tensile tests; the final value of the multi-criterion cost function MCF was calculated for each combination of the Cowper–Symonds parameters:

_{LSR,k}cost function was divided by 1,000,000 to put it on the same order of magnitude as the cost function SSQD

_{HSR,k}. Many different values of the mixing weight u, ranging from u = 0.2 to u = 0.8. were tested. It turned out that the final estimates of the most optimal parameters C and p are quite insensitive to the value of u. For this reason, the value of u = 0.5 was selected for further analysis.

## 3. Results

#### 3.1. Tensile Stress-Strain Curves at Different Low-Strain Rates

_{p0.2}) at the lowest strain rate and the ultimate tensile strength (R

_{m}) from the tensile tests were approximately 800 MPa and 1100 MPa, respectively.

_{m}of the material is reached, the true stress and true strain values should be calculated using the measured specimen cross-section at a specified sampling time. Along with the measured force at the same time, the true stress and true strain were calculated as follows:

_{0}is an initial cross-section, and A and F are measured cross-section and force, respectively. Since it is difficult to perform the second part of the transformation manually during the experiment, the final true stress was determined only for the ruptured specimen, while the true strains were measured throughout the whole experiment using the high-speed camera combined with the DIC method. To estimate the true width of the specimen during the test, a tracking algorithm in ProAnalyst V1.8 software from Xcitex, Woburn, MA, USA, was applied.

^{−1}and 5.4 s

^{−1}, the use of the mechanical extensometer was not a viable option. Consequently, only the strain data, which was determined with the DIC method, was available for these strain rates. For the lower strain rates, the strains from the DIC method were compared to the final strain from Equation (16) for the ruptured specimens as well as to the transformed values according to Equation (14) before the engineering tensile strength R

_{m}was reached. It turned out that true strains were correctly determined using the DIC method. The obtained true stress-true strain curves are shown in Figure 12.

#### 3.2. High Strain-Rate Experiments with a Shooting Ball Test

#### 3.3. Estimating the Cowper–Symonds Parameters from the Experiments

_{LSR}/1,000,000 for SIGY = 400 MPa and the ranges of the C and p parameters in Table 2 are presented in Figure 14.

_{LSR}in Figure 14. The best two solutions among them (for differently high quasi-static flow stress curves, of course) were the following: (SIGY = 500 MPa, C = 300,000 s

^{−1}, p = 7) and (SIGY = 400 MPa, C = 210 s

^{−1}, p = 30). The corresponding cost function values, according to Equation (4), were almost identical (i.e., only 5% different). The first candidate solution was much closer to the values from the literature, as presented in Section 1. Then, we performed a rough reverse-engineering approach (i.e., a simplified version of Phase 2) for both candidate solutions using the LS-Dyna simulations. It turned out that the second candidate completely outperformed the first (more logical) one. Its cost function, according to Equation (13), was 30% smaller.

^{−1}and $450{\mathrm{s}}^{-1}$. For the first optimum candidate ($SIGY=500\mathrm{MPa},C=300,000{\mathrm{s}}^{-1},p=7$), it can be seen that the corresponding Cowper–Symonds term ${\left({\dot{\epsilon}}_{pl}/C\right)}^{1/p}$ is equal to 0.0015, which is very close to zero. This means that even a small variation in the strain rate results in a significant change of the term ${\left({\dot{\epsilon}}_{pl}/C\right)}^{1/p}$. In contrast, the second optimum candidate ($SIGY=400\mathrm{MPa},C=210{\mathrm{s}}^{-1},p=30$) is located to right of the (1,1) point in Figure 15. This means that the response of the term ${\left({\dot{\epsilon}}_{pl}/C\right)}^{1/p}$ to the variations of the strain rate is moderate and not abrupt, which is phenomenologically correct. Therefore, the parameter SIGY was kept constant at 400 MPa in Phase 2, because this was chosen as the most appropriate value in our case.

^{−1}and p = 34.6. If Figure 14, Figure 16, and Figure 17 are compared, it can be concluded that the SSQDHSF cost function actually determines the final optimal values of the C and p parameters due to the large flat plateau of the cost function SSQDLSR. A comparison with the results from the literature is presented in Section 4.

## 4. Discussion

^{−1}and p = 34.6) to these four reliable estimates from the literature, we can conclude that our estimates are reasonable for the available experimental data.

- Most scholars applied the conventional formulation for the MAT_24 material, as shown in Equation (2), despite the fact that such a formulation is sometimes not appropriate for the studied steel alloys.
- Furthermore, it is not optimal if the parameters C and p are estimated only on the basis of the high-strain rate experiments (e.g., Taylor test, split Hopkinson bar test, etc.). In the past [49], we also followed such an approach and obtained similar parameters. However, as soon as the optimisation process is extended to a wide range of strain rates, it turns out that the parameter values of C > 100,000 are not the best estimates. Despite the fact that different scholars have performed tests at different strain rates it is not clear how they considered the experimental data at various strain rates when estimating the parameters C and p.

## 5. Conclusions

- The obtained results were compared to the values from the literature. It turned out that the estimated optimal values of the parameters C and p sometimes deviate significantly from the literature. After a careful analysis of our results and the published results from the literature, it can be concluded that some estimates in the literature are not very reliable due to different reasons that are explained in the article. This critical evaluation of our results and references presents another novelty of this article.
- Yet another result of this study is a conclusion that it is immensely important is how the cost function is defined, which measures a deviation between the experimental results and numerical simulations for the high strain-rate experiments. It turns out that the integral measure of deviation is much better than the point-wise measure of deviation.
- Finally, we can also conclude that the high-strength steel SZBS800 is not very sensitive to the strain, which is indicated by a rather high value of the Cowper–Symonds parameter p. However, such results are not surprising because it was previously discovered by other scholars that high-strength steels are often insensitive to the high strain-rate effects.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flat specimen for tensile tests according to ASTM E8/E8M standard (Reprinted from [43]) (all presented dimensions are in mm).

**Figure 2.**Experimental setup for measuring the deformed geometry of the specimens with the Dantec 3D DIC system.

**Figure 5.**Used flow curve ${\sigma}_{Y,S}\left({\epsilon}_{pl}\right)$ at quasi-static strain rates.

**Figure 7.**Results obtained from impact simulations with LS-Dyna software. Colour bar represents the displacement values along y direction. (

**a**) Results at initial timestamp before impact. (

**b**) Results at final timestamp after impact.

**Figure 8.**Reference images for different image resolutions. (

**a**) Using 2-times higher resolution than the original simulation. Coordinate increment was $dx=dz=0.3125$ with matrix numbers of columns and rows, respectively ${n}_{x}=192$, ${n}_{z}=208$. (

**b**) Using the original simulation resolution. Coordinate increment was $dx=dz=0.625$ with matrix numbers of columns and rows, respectively ${n}_{x}=96$, ${n}_{z}=104$. (

**c**) Using 2-times lower resolution than the original simulation. Coordinate increment was $dx=dz=1.25$ with matrix numbers of columns and rows, respectively ${n}_{x}=48$, ${n}_{z}=59$. The colour scale is the same as in the Figure 7 and represents the displacement values along y direction.

**Figure 9.**DIC references images for different image resolutions. (

**a**) Using a 2-times higher resolution than the original DIC mesh size. Coordinate increment was $dx=dz=0.25$ with matrix numbers of columns and rows, respectively ${n}_{x}=333$, ${n}_{z}=466$. (

**b**) Using the original simulation resolution as DIC mesh size. Coordinate increment was $dx=dz=0.5$ with matrix numbers of columns and rows, respectively ${n}_{x}=167$, ${n}_{z}=233$. (

**c**) Using a 2-times lower resolution than the original DIC mesh size. Coordinate increment was $dx=dz=1$ with matrix numbers of columns and rows, respectively ${n}_{x}=84$, ${n}_{z}=117$. The colour scale is the same as in the Figure 7 and represents the displacement values along y direction.

**Figure 10.**DIC reference images with cropping rectangles. (

**a**) DIC reference image with 2-times higher resolution. (

**b**) DIC reference image with original resolution. (

**c**) DIC reference image with 2-times lower resolution. The colour scale is the same as in the Figure 7 and represents the displacement values along y direction.

**Table 1.**Chemical composition of SZBS800 steel (Adapted from [42]).

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% |

Parameter | Values |
---|---|

C [s^{−1}] | 10, 30, 50, 70, 90, 1110, 130, 150, 170, 190, 210, 230, 250, 270, 300, 600 |

p [/] | 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100 |

Parameter | Values |
---|---|

C [s^{−1}] | 70, 150, 230, 310, 390, 470, 550, 630, 710, 790, 870, 950, 1030, 1110, 1190, 1270, 1350 |

p [/] | 15, 20, 25, 30, 35, 40, 45 |

Parameter | Values |
---|---|

Impact angles α [°] | 0, 0, 0, 20, 20, 20, 35, 35, 35 |

Impact velocity v [m/s] | 111.5, 133.3, 155.3, 111.2, 132.6, 154.2, 113.1, 131.9, 153.5 |

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**MDPI and ACS Style**

Škrlec, A.; Panić, B.; Nagode, M.; Klemenc, J.
Estimating the Cowper–Symonds Parameters for High-Strength Steel Using DIC Combined with Integral Measures of Deviation. *Metals* **2024**, *14*, 992.
https://doi.org/10.3390/met14090992

**AMA Style**

Škrlec A, Panić B, Nagode M, Klemenc J.
Estimating the Cowper–Symonds Parameters for High-Strength Steel Using DIC Combined with Integral Measures of Deviation. *Metals*. 2024; 14(9):992.
https://doi.org/10.3390/met14090992

**Chicago/Turabian Style**

Škrlec, Andrej, Branislav Panić, Marko Nagode, and Jernej Klemenc.
2024. "Estimating the Cowper–Symonds Parameters for High-Strength Steel Using DIC Combined with Integral Measures of Deviation" *Metals* 14, no. 9: 992.
https://doi.org/10.3390/met14090992