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Research Papers

# Flow Stress and Damage Behavior of C45 Steel Over a Range of Temperatures and Loading RatesOPEN ACCESS

[+] Author and Article Information
Farid H. Abed

Department of Civil Engineering,
American University of Sharjah,
SHJ 26666, United Arab Emirates
e-mail: fabed@aus.edu

Department of Civil Engineering,
American University of Sharjah,
SHJ 26666, United Arab Emirates

Akrum Abdul-Latif

Département GIM 3 rue de la Râperie,
Université Paris 8 IUT de Tremblay-en-France,
Tremblay-en-France, France

Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70802

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 9, 2016; final manuscript received September 9, 2016; published online February 7, 2017. Assoc. Editor: Taehyo Park.

J. Eng. Mater. Technol 139(2), 021012 (Feb 07, 2017) (8 pages) Paper No: MATS-16-1175; doi: 10.1115/1.4035488 History: Received June 09, 2016; Revised September 09, 2016

## Abstract

This research aims to describe the behavior of C45 steel and provide better understanding of the thermomechanical ductile failure that occurs due to accumulation of microcracks and voids along with plastic deformation to enable proper structural design, and hence provide better serviceability. A series of quasi-static tensile tests are conducted on C45 steel at a range of temperatures between 298 K and 923 K for strain rates up to 0.15 s−1. Drop hammer dynamic tests are also performed considering different masses and heights to study the material response at higher strain rates. Scanning electron microscopy (SEM) images are taken to quantify the density of microcracks and voids of each fractured specimens, which are needed to define the evolution of internal defects using an energy-based damage model. The coupling effect of damage and plasticity is incorporated to accurately define the constitutive relation that can simulate the different structural responses of this material. Good correlation was observed between the proposed model predictions and experiments particularly at regions where dynamic strain aging (DSA) is not present.

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## Introduction

Many research studies and experiments have been carried out in the past few decades to study the material behavior at elevated temperatures and strain rates. Steel, in particular, is one of the materials that have been extensively studied for that purpose due to its wide use in the industry. Researchers have been trying to understand and quantify experimentally the parameters which describe the thermomechanical behavior of steel [18] including its damage behavior [912], and numerically by developing constitutive equations to describe the flow stress of different types of mild and high strength steel [1323]. The interest of this research is to study the thermomechanical behavior of C45 steel at elevated temperatures and quasi-static strain rates.

Steel suppliers are providing C45 mainly to build tools, machines, or structures that sustain impact loading, sharp cutting edges, corrosion, brittle fracture, and to provide high performance and weldability. Also, this type of steel is known for its wear resistance and ability for quenching and tempering. Many researchers have tried in the past decades to carry out experiments and analysis to study different perspectives of C45 steel [2434]. These studies were attempting to understand the behavior of C45 steel in terms of formability, effect of different kinds of processes such as heat treatment and cooling, Nanostructural behavior under impact loading, distortion under cooling effect, behavior of the material when exposed to lubrication and corrosion protection processes, vibration effect, friction and wear properties, surface roughness, and machinability.

However, the literature lacks research related to investigating the flow stress of C45 steel and its damage evolution. The currently increasing use of this type of steel as structural element in the offshore structures necessitates the need to study its constitutive behavior for design purposes. This research aims to fill this gap by conducting experimental tests to investigate the stress–strain behavior of C45 steel under different combinations of temperatures and strain rates. In addition, SEM images are utilized to quantify the density of microcracks and voids at fracture. The SEM results will be utilized in the evolution of microcracks and voids throughout the inelastic deformation using an energy-based damage model [9].

## Experimental Tests Results and Discussion

The present experimental program considers quasi-static tests at room and high temperatures, drop-hammer tests at room temperatures, and SEM images at fracture. The material specimens were manufactured from thick structural C45 steel plate. The chemical composition of C45 steel used in this study is shown in Table 1.

###### Quasi-Static Tests at Room and High Temperatures.

Tensile coupon tests at room temperature and three different strain rates of 0.0015, 0.015, and 0.15 s−1 were conducted using a universal testing machine (UTM) with 100 kN capacity. Two specimens with the geometry shown in Fig. 1 were tested for each case.

The measured stresses and strains were recorded and converted into true stresses and true strains as presented in Fig. 2.

As shown in Fig. 2, the yield stresses along with the ultimate stresses are increasing with the increase of the strain rate, which is a typical trend of steel. It can also be noticed that strain hardening trend is similar regardless of the loading rate and the main difference is the yield point. In other words, the stress–strain curves will be almost identical if the yield point is the same. The increase in yield stress is interpreted physically as the resistance of initial dislocation by the Peierls barriers provided by the lattice itself [16].

The quasi-static tensile tests were extended to investigate the mechanical response of C45 steel at higher temperatures. For that purpose, three temperatures of 523 K, 723 K, and 923 K at two quasi-static loading rates were considered. The thermal tests were conducted using a UTM coupled with a heat chamber to apply the needed temperatures. Two rounded specimens with the geometry shown in Fig. 3 were tested for each thermal test.

The true stress–true strain curves for these high temperature tests are shown in Figs. 4 and 5 for strain rates of 0.0015 s−1 and 0.15 s−1, respectively.

It can be noticed from Fig. 4 for strain rate of 0.0015 s−1 that the stresses are generally decreasing with the increase of temperature. The difference in the strain hardening between temperatures of 523 K and 723 K is increasing with strain. Also, the plastic region at temperature of 923 K is almost flat, which indicates that the strain hardening at this high temperature level is diminished, and the remaining are denoted as athermal stresses [16]

The same observations can be made on the true stress–true strain curves shown in Fig. 5 for thermal tensile tests performed at strain rate of 0.15 s−1. However, it is clearly shown that the stresses for the specimen at temperature of 723 K are higher than the stresses at 523 K. Such a jump in the stress value is attributed to what is called the dynamic strain aging (DSA) phenomenon that is usually encountered in high strength ferrite and austenite steel and at certain combinations of temperatures and strain rates.

It can be deduced from the above stress–strain results that the thermal flow stress of C45 steel is mainly controlled by the yield stress for the range of quasi-static rates considered. The strain hardening, on the other hand, is mainly related to the athermal part of the flow stress at regions where the DSA is inactive. Such behavior is similar to the flow stress of most of bcc metals and ferrite steel [16].

###### Dynamic Strain Aging.

According to Wang et al. [35], dynamic strain aging is basically a phenomenon that occurs due to the interaction between “diffusing solute atoms” and “mobile dislocations” in the material. It usually occurs after “critical strain” at specific “regime” of strain rate and temperature. The authors stated that the range of temperatures at which this phenomenon occurs is between 20% and 50% of the melting temperature. The effect of DSA can be expressed in the material as “serrated flow,” “negative strain rate sensitivity,” and “peaks or plateaus on flow stress/temperature plots.”

For C45 steel, the DSA behavior can be clearly noticed when the stress variations with temperature are plotted for different strain levels as shown in Figs. 6 and 7 for the lower and higher strain rates considered. The peaks/jumps in the flow stress are clearly noticed at temperature of 523 K and partially at temperature 723 K for strain rate of 0.0015 s−1 and at temperature of 723 K for strain rate of 0.15 s−1.

###### Drop Hammer Test.

The drop hammer test is used to determine the dynamic behavior of a sample at a strain rate in the order of . During the test, a mass falls from a given height on a sample transferring the potential energy in the mass into kinetic energy in the sample. Two cylindrical specimens with a diameter of 8 mm and a height of 16 mm (Fig. 8) were tested at room temperature using two different hammer masses of 28 kg and 56 kg dropped from heights of 4 m and 2 m, respectively. This was done to ensure the same energy for two loading rates equivalent to 390 s−1 and 550 s−1.

The drop hammer tests are carried out using a dynamic load cell of 20 ton and a laser beam displacement transducer as well as a 5000 g accelerometer. These instruments are connected to an acquisition chain that ensures the simultaneous recording of force, displacement, and acceleration. In addition, two photocells are used to assure the synchronization of the acquisition of these essential data. The position of these photocells represents one of the principal difficulties related to the synchronization of the collected experimental data. The data recording needed filtering especially the impact force. The inertial effects are assumed negligible since the impact velocities are relatively low (lower than 30 m/s) [36]. A filtering process regarding the impact force is achieved using the so-called Chebyshev filter. Therefore, the axial stress and strain are computed via the filtered impact force and displacement, and converted into true stress and true strain, respectively, as shown in Fig. 9.

It is difficult to identify a clear point, which defines the yield of the material and its flow stress behavior at such a test setup. This can be clearly noticed in Fig. 9, where the curves are increased until they achieve the ultimate stress and then start to rebound. The main difference between both loading scenarios is the ultimate point achieved in which the ultimate stresses are increased with the increase of the strain rate.

###### Scanning Electron Microscopy.

This part of the experimental program is intended to characterize the material damage by processing the SEM images to estimate the density of microcracks at fracture. For each loading condition, the fractured surface of each specimen after each test was polished using a grinder polish machine. A cut through the sample is needed to get clear fractured surfaces taking into consideration that the cut should be performed slowly in order not to heat the sample and to get a clear fractured surface without changing the mechanical property of the sample, erasing the cracks inside the sample, or even causing any further cracks [37]. Figure 10 presents a sample of SEM images for fractured specimens at a strain rate of 0.15 s−1 and various temperatures.

The quantification results for the material damage in terms of density of microcracks and voids are presented in Fig. 11. The density of cracks and voids was calculated by analyzing the SEM images by detecting the contrast between the pixels, which contain cracks and the pixels that do not contain cracks. In order to obtain a good average crack density, five to seven different areas on the fractured surface were selected and analyzed [37]. The results presented in Fig. 11 clearly explain the increasing pattern of the damage values with the increase of temperature and strain rates. These results are utilized to identify the material constants for an energy-based damage model as will be discussed in Sec. 3.

## Constitutive Modeling

A coupled damage-plasticity constitutive modeling was considered in this study to describe the flow stress of C45 steel. This was done by integrating the energy-based damage model into the Johnson–Cook (JC) model to accurately capture the stress–strain behavior of C45 steel. The concept of effective stress, which relates the undamaged and damaged state of the material, was used considering isotropic damage variable ϕ as shown in Eq. (1). In this case, the effective flow stress was calculated using the empirical relationship of JC model, and the evolution of damage will be determined according to Abed et al. [9]. Display Formula

(1)

where $σ¯$ is the effective stress in the body which is calculated using the JC relationship, $ϕ$ is the damage factor, and $σ$ is the stress of the damaged body.

###### Energy-Based Damage Evolution Model.

As explained earlier, Abed et al. [9] developed an energy-based model that is capable of describing the damage evolution in steel using the principles of continuum damage mechanics. The model describes the damage as the increase in the dissipated energy of the material. The model is expressed as a function of the dissipated energy ratio to the total dissipated energy and the damage at fracture as shown in the below equation Display Formula

(2)

where $ϕ$ is the damage at the point of interest during the deformation process, $ϕf$ is the damage at fracture which is measured using SEM images (Fig. 10), $Up$ is the corresponding dissipated energy at this point of interest, $UPT$ is the total dissipated energy, and the exponent $α$ is a constant that determines the trend of damage evolution throughout the deformation process. Since most steel tend to have similar damage evolution trend, the value of the exponent $α$ for C45 steel is assumed equal to 2.0 similar to the one obtained by Abed et al. [9].

The dissipated plastic energy can be determined using Eq. (3), where σ and ε are the true stress and true strain, respectively. Display Formula

(3)

Figure 12 presents the damage evolution results for C45 steel for all loading combinations considered. The damage grows with the increase of strain rate except for the case at temperature of 723 K. This could be attributed to the strain aging phenomenon encountered that leads the material to gain more strength at this loading combination of temperature and strain rate. Also, the strain aging can be noticed affecting the damage evolution at a temperature of 523 K. Overall, the results seem to be reasonable and they seem to follow the trend that the high strength steel follows at these conditions [8].

It can also be noticed from Fig. 12 that the damage effect starts to grow on higher scale at a certain strain point for each case. At the beginning, the damage evolution is almost constant until a certain strain point at which the damage starts to grow faster, leading to the degradation of the material. This is because the dissipated plastic energy, which is represented by the area under the stress–strain curve, is increasing with strain until it reaches a value of high ratio compared to the total energy. At that point, the damage effect starts to be noticed clearly.

###### Flow Stress Using JC Model.

Many flow stress models have been developed to describe the materials nonlinear behavior when exposed to high strain rates and temperatures. Some are empirical or semi-empirical relations, while others are physically based sharing the same purpose of describing the change in mechanical properties of materials when they experience the accumulation of plastic deformation. They serve the purpose of producing the material true stress–true strain curves under different loading conditions. These modeling concepts come from the fact that the mechanical response depends on the rate of deformation as well as the evolving temperature.

In this paper, the JC model, which was proposed by Johnson and Cook in 1983 [23], is considered to predict the flow stress of C45 steel. It is an empirical relation that describes the flow stress in terms of strain, strain rate, and temperature as given by the below equation Display Formula

(4)

where $σ$ is the equivalent stress, $εp$ is the plastic strain, $έ$ is the strain rate, $έo$ is the reference strain rate, $T$ is the temperature of the sample, $Tmelt$ is the melting temperature of the material, $Tref$ is the reference temperature, A is the yield stress of the material at a reference deformation conditions, B is the strain hardening constant, C is the strain rate strengthening coefficient, n is the strain hardening constant, and m is the thermal softening coefficient. The JC material constants for C45 steel obtained using the experimental results presented earlier are listed in Table 2.

In general, the JC model shows acceptable comparisons at certain strain values and realistically captures the nonlinear trend of C45 steel over the range of temperatures and strain rates considered. However, some deviation can be noticed in regions where the strain aging is active. Figure 13 shows a very good comparison between the stress–strain results predicted by the JC model and experiments at room temperature. The coupled JC-damage model was capable of predicting the values and trends of the stresses up to fracture for the three different loading rates. On the other hand, the model was unable to capture the dynamic strain aging phenomenon encountered at temperatures of 523 K and 723 K as clearly shown in Figs. 14 and 15. As mentioned earlier, dynamic strain aging phenomenon is a very complicated process; therefore, a physical-based constitutive term that is able to capture this increase in strength at the microstructural level is required. The DSA effect becomes active when the aging time is equal to the waiting time of mobile dislocations, which in turn increases the obstacle strength by certain amount (i.e., jump) [15]. Therefore, in order to predict this stress increase, an additional mathematical expression involving the DSA physical characterization is required. For this reason, it is fair to say that the C45 material constants obtained in this study for the JC model are more applicable at room temperature conditions.

###### Effect of Damage.

This section is intended to show an example of the effect of including the energy-based damage evolution on the overall stress–strain response by direct comparison of the experimental results with JC model prediction with and without including the damage evolution. The comparison was conducted for loading scenarios where the model captured the behavior almost successfully; i.e., not affected by the dynamic strain aging phenomenon. Hence, loading scenarios at room temperature and 923 K for loading rates of 0.0015 s−1 and 0.15 s−1 were considered in this comparison as shown in Fig. 16. The integration of the energy-based damage model into the JC equation succeeded in capturing the material softening behavior.

## Conclusion

The C45 steel is increasingly being used as structural material in the oil and gas industry. The material response to typical loading conditions experienced in such environment needs to be tested and evaluated. Hence, a comprehensive assessment of the thermomechanical response of this type of high strength steel is timely and necessary for design purposes.

This research studied the thermomechanical behavior of C45 structural steel based on the results of a series of experimental tests that were conducted at different levels of temperatures (298–923 K) at strain rates. The corresponding true stress–true strain results were utilized to identify the material constants for constitutive modeling, which is required to understand quantitatively the deformation behavior of this material. SEM was utilized to understand the microstructure of the material and measure the internal microcracks and voids for C45 steel at fracture for each test condition. The SEM results were then implemented in an energy-based damage model to define the evolution of damage throughout the deformation process. The damage model was integrated into the JC plasticity model to describe the flow stress of C45 steel. The following conclusions can be drawn:

1. The quasi-static tests results at high temperatures revealed the typical degradation trend of the thermal stresses with temperature increase. However, DSA took place at certain combination of temperatures and strain rates. The DSA is usually characterized by the sudden jump in the thermal stress variation with temperature. This DSA phenomenon was clearly observed at 523 K for the case of 0.0015 s−1 strain rate and at 723 K for the higher rate.
2. The yield stress increased with the increase of strain rate and decreased with temperature except in the regions where the DSA is active. On the other hand, the strain hardening was found to be almost rate-independent at room temperature. However, the rate dependency behavior of the material increased with temperature.
3. A constitutive description of C45 steel is required to simulate the flow stress of the material. In this research, the empirical relation of JC constitutive model was selected to describe the material response. The DSA effects, however, were not captured by this model.
4. The energy-based damage model captured the damage evolution within the microstructure of the material and the failure mode beyond the ultimate point.

The application of this research outcome is limited to the loading conditions considered in the present experimental program to get accurate results. Any other scenarios intended to be implemented shall consider further experimental tests and exploration. Also, the DSA phenomenon should be studied further in order to understand its effect on the material response. To capture such effect, a physical-based constitutive model should be developed to accurately describe the plasticity behavior of C45 steel. The model should be able not only to address the plastic deformation that is attributed to the motion of dislocations but also to include the plastic flow in the range of temperatures and strain rates where diffusion and creep are dominant, i.e., the regions where DSA is active.

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## References

Nemat-Nasser, S. , and Guo, W. , 2005, “ Thermomechanical Response of HSLA-65 Steel Plates: Experimental and Modeling,” Mech. Mater., 37(5), pp. 379–405.
Celentano, D. J. , and Chaboche, J.-L. , 2007, “ Experimental and Numerical Characterization of Damage Evolution in Steels,” Int. J. Plast., 23(10–11), pp. 1739–1762.
Nemat-Nasser, S. , and Guo, W. , 2003, “ Thermomechanical Response of DH-36 Structural Steel Over a Wide Range of Strain Rates and Temperatures,” Mech. Mater., 35(11), pp. 1023–1047.
Guo, W.-G. , and Nemat-Nasser, S. , 2006, “ Flow Stress of Nitronic-50 Stainless Steel Over a Wide Range of Strain Rates and Temperatures,” Mech. Mater., 38(11), pp. 1090–1103.
Nemat-Nasser, S. , Guo, W.-G. , and Kihl, D. P. , 2001, “ Thermomechanical Response of AL-6XN Stainless Steel Over a Wide Range of Strain Rates and Temperatures,” J. Mech. Phys. Solids, 49(8), pp. 1823–1846.
Su, J. , Guo, W. , Meng, W. , and Wang, J. , 2013, “ Plastic Behavior and Constitutive Relations of DH-36 Steel Over a Wide Spectrum of Strain Rates and Temperatures Under Tension,” Mech. Mater., 65, pp. 76–87.
Vaynman, S. , Fine, M. E. , Lee, S. , and Espinosa, H. D. , 2006, “ Effect of Strain Rate and Temperature on Mechanical Properties and Fracture Mode of High Strength Precipitation Hardened Ferritic Steels,” Scr. Mater., 55(4), pp. 351–354.
Rohr, I. , Nahme, H. , and Thoma, K. , 2005, “ Material Characterization and Constitutive Modelling of Ductile High Strength Steel for a Wide Range of Strain Rates,” Int. J. Impact Eng., 31(4), pp. 401–433.
Abed, F. H. , Al-Tamimi, A. K. , and Al-Himairee, R. M. , 2012, “ Characterization and Modeling of Ductile Damage in Structural Steel at Low and Intermediate Strain Rates,” J. Eng. Mech., 138(9), pp. 1186–1194.
Darras, B. , Abed, F. , Pervaiz, S. , and Abdu-Latif, A. , 2013, “ Analysis of Damage in 5083 Aluminum Alloy Deformed at Different Strain rates,” Mater. Sci. Eng.: A, 568, pp. 143–149.
Chae, D. , and Koss, D. A. , 2004, “ Damage Accumulation and Failure of HSLA-100 Steel,” Mater. Sci. Eng.: A, 366(2), pp. 299–309.
Al-Himairee, R. M. , Abed, F. H. , and Al-Tamimi, A. K. , 2011, “ Damage Evolution in Structural Steel at Different Loading Conditions,” Key Eng. Mater., 471–472, pp. 969–974.
Abed, F. , and Makarem, F. , 2012, “ Comparisons of Constitutive Models for Steel Over a Wide Range of Temperatures and Strain Rates,” ASME J. Eng. Mater. Technol., 134(2), p. 021001.
Goto, D. , Garrett, R. , Bingert, J. , Chen, S. , and Gray, G. , 2000, “ The Mechanical Threshold Stress Constitutive-Strength Model Description of HY-100 Steel,” Metall. Mater. Trans. A, 31(8), pp. 1985–1996.
Abed, F. , Ranganathan, S. , and Serry, M. , 2014, “ Constitutive Modeling of Nitrogen-Alloyed Austenitic Stainless Steel at Low and High Strain Rates and Temperatures,” Mech. Mater., 77, pp. 142–157.
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## Figures

Fig. 1

Geometry of the tensile test specimens at room temperature with an average width w = 12 mm, thickness t = 10 mm, and gauge length L = 57 mm

Fig. 2

True stress–true strain curves at room temperature for different strain rates

Fig. 3

Geometry of the tensile test specimens at high temperatures with an average diameter of 6 mm and gauge length of 39 mm

Fig. 4

True stress–true strain curves at high temperatures for strain rate of 0.0015 s−1

Fig. 5

True stress–true strain curves at high temperatures for strain rate of 0.15 s−1

Fig. 6

Stress variations against temperature for strain rate 0.0015 s−1 at various strain levels

Fig. 7

Stress variation against temperatures for strain rate 0.15 s−1 at various strain levels

Fig. 8

Geometry of the drop hammer test specimens

Fig. 9

True stress–true strain curves for drop hammer test at strain rates of 390 s−1 and 550 s−1

Fig. 10

SEM images of fractured damaged surfaces at different temperatures for strain rate of 0.15 s−1, (a) T = 298 K, (b) T = 523 K, (c) T = 723 K, and (d) T = 923 K

Fig. 11

Damage Ф at fracture for each rate of deformation at all temperatures

Fig. 12

Damage ϕ evolution in the material for different strain rates and temperatures

Fig. 13

Stress–strain curves at T = 298 K for strain rates of (a) 0.0015 s−1, (b) 0.015 s−1, and (c) 0.15 s−1

Fig. 14

Stress–strain curves for strain rate 0.0015 s−1 at (a) T = 523 K, (b) T = 723 K, and (c) T = 923 K

Fig. 15

Stress–strain curves for strain rate 0.15 s−1 at (a) T = 523 K, (b) T = 723 K, and (c) T = 923 K

Fig. 16

Comparison between experimental results, JC model, and coupled plasticity-damageJC model for (a) strain rate = 0.0015 s−1 at T = 298 K, (b) strain rate = 0.15 s−1 at T = 298 K, (c) strain rate = 0.0015 s−1 at T = 923 K, and (d) strain rate = 0.15 s−1 at T = 923 K

## Tables

Table 1 Chemical composition of C45 steel
Table 2 JC parameters

## Discussions

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