A modified Johnson-Cook model for dynamic behavior of spray-deposition 17 vol.% SiCp/7055Al composites at high strain rates

In this study, the dynamic impact tests of spray-deposited 17 vol% SiCp/7055Al composites at various strain rates were performed with a Split Hopkinson Pressure Bar (SHPB). In these tests, the strain rate was 392 s−1–2002 s−1, and the temperature was 293 K–623 K. Subsequently, the Johnson-Cook (JC) was used to describe the flow behaviors under high speed impact deformation, and its effectiveness was assessed. Results show that the stress values predicted by the JC model could be inconsistent with the experimental ones. A modified JC constitutive model of 17 vol% SiCp/7055Al composites was developed by modifying the strain rate hardening term and considering coupling effects of strain, temperature and strain rate. According to the comparison between the experimental data and the results assessed with the modified JC model, the proposed model could assess the stress-strain values more accurately, especially in the beginning of plastic deformation. This indicates that the composites exert the joint effects of strain rate hardening and temperature softening during high-speed impact deformation.

In recent decades, domestic and foreign scholars have fabricated SiCp/Al composites based on spraydeposition and conduct relevant research [11][12][13]. However, existing researches mainly focus on mechanical properties, the interface effects on the mechanical properties and static deformation characteristics [14,15]. As a matter of fact, composite components are likely to experience dynamic impact loading in several applications, it is generally known that all materials exhibit different deformation characteristics under static and dynamic loading conditions [1,5]. Studying flow behaviors of SiCp/Al composites at high strain rates is critical to explain the dynamic characteristics of the material in their application [16]. However, the dynamic impact test can obtain the performance parameters and dynamic flow characteristics of the spray-deposition SiCp/Al composites at high strain rates.
Under the varied loading, flow behaviors of the materials were affected by the strain, strain rate as well as temperature. On the whole, the flow behaviors of materials during hot deformation are complex, whereas the constitutive relationship can describe the stress-strain relationship of materials in a mathematical model [16][17][18]. The applicable constitutive model should be capable of expressing the dynamic characteristics of the materials under various loading conditions, which is a prerequisite for accurate numerical analysis of material deformation including finite element simulation [17]. Constitutive equations of materials have been primarily split into two categories [19]: Physically-based constitutive models and phenomenological constitutive models. As compared with physics-based models, however, phenomenological constitutive models involve fewer material constants and require limited experimental data; they are always prioritized by the users to assess the stress-strain values of materials. Besides, the phenomenological constitutive models have been successfully adopted to describe the sophisticated flow behaviors of materials under larger loading forming conditions [20,21].
The Johnson-Cook (JC) constitutive model has been extensively employed in phenomenological constitutive models for its simple form and simplified calculation [21][22][23]. The JC model initially proposed by Johnson and Cook in 1983, has been adopted for large deformation, high strain rates and high temperature of metals [23][24][25]. The JC model considering strain rate hardening, strain hardening and thermal effect; it can describe the flow behaviors of various materials under specific loading conditions. It is noteworthy that the JC model has been extensively used in impact dynamics research [26]. The original JC model only gives the expression of yield stress, and the material constants are easy to acquire from experimental. However, it is found that the original JC model has some deviations in the prediction of deformation behavior. To enhance the accuracy of the JC model, the model also has been widely modified in available literature. Among these proposed modified JC models, strain, strain hardening and temperature softening terms in the modified models have been applied most frequently. These researches primarily established JC models of alloys and partial composites. At present, the spray-deposition processed SiCp/Al composites have been rarely reported, let alone constructing the JC constitutive equation of spray-deposition SiCp/7055Al composites. Thus far, whether the existing related constitutive model is applicable to the SiCp/7055Al composites remains unclear.
In the present work, dynamic uniaxial compression tests were performed on the spray-deposited 17 vol% SiCp/7055Al composites, the stress-strain data were obtained with a Split Hopkinson Pressure Bar (SPHB); besides, their effects on the flow behaviors were discussed. The JC model and the modified JC model constitutive equation of SiCp/7055Al composites were constructed to describe the dynamic behavior. The deformation behaviors of SiCp/7055Al composites under various strain rates at different temperatures were discussed. As revealed from the results, the predicted values of the original JC equation are greatly different from the experimental values, and the proposed modified JC equation is capable of precisely assessing the composites deformation behaviors.

Experimental material
In the present study, the commercial 7055 aluminum alloy was used. 7055 aluminum alloy have been widely used in aerospace, transportation and other fields because of its excellent mechanical properties, its chemical composition is given in table 1. SiC particles with the size of 15-20 μm acted as the reinforcement. The SiCp/ 7055Al composites were fabricated by spray-deposition. Before spray-deposition, to reduce the agglomeration of SiC particles, the SiC particles were heated for 10 h at 523 k to remove crystalline water and adsorbents. The spray-deposition process parameters included: Atomization temperature at 1023-1123 K; Nebulizer pressure under 0.6-0.8 MPa; The diameter of sedimentary disk as 530 mm; Matrix rotation speed at 150-250 r min −1 ; Powder-feeding pressure under 0.1-0.2 MPa. The volume fraction of added alpha-SiC particles was 17%, and the density of the composite was 92.3%. The size of the deposited cylindrical sample is 160×320 mm.

Experience of dynamic tests
The cylindrical impact sample size is 10 mm in diameter and 6 mm in height. The cylindrical bar specimens processed from the original spray-deposition 17 vol% SiCp/7055Al composites with EDM PW2UP wire cutter. And all the test specimens were processed to develop the coincident axis along the radial direction to ensure consistency. SHPB was adopted to perform the dynamic compressive tests, and the incident, reflected and transmitted waves were transmitted to the data processing system by using the strain gauge on the incident bar and the transmission bar. The strain gauge resistance of SHPB's data acquisition system is 120 Ω. To minimize the impact of shock waves on the results of the experiment, a small amount of vaseline was applied on both ends of the sample and 2 mm long rubber was applied on the other end of the incident bar. In accordance with the one-dimensional stress wave theory, the strain rate ( e), strain (ε) and stress (σ) of the tested material can be Where C and E are the elastic wave velocity and the Young's modulus in the bars, respectively.
are the incident wave amplitude, the reflected wave amplitude and the transmitted wave amplitude, respectively; l 0 is the initial length of the specimen; A, A 0 are the bar cross-sectional area and the specimen crosssectional area.
The dynamic test stress-strain curves can be obtained by eliminating the time term. The specimens were subjected to high-speed impact tests on SPHB at the strain rate ranging from 392 s −1 to 2002 s −1 , as well as at the temperatures of 293 K, 523 K, 573 K and 623 K, respectively.   indicates that the precipitate of the materials phase is Al 2 Cu. In the meantime, some weak peaks of magnesium compounds (such as Al 2 CuMg and MgZn 2 ) were also found in the XRD pattern, but these could not be completely determined to be Al 2 CuMg and MgZn 2 . This may be due to the low content of Mg compounds precipitates in the spray-deposition composites as shown in figures 1(c) and (d).

Flow behavior
The strain rate ( e), strain (ε) and stress (σ) data of the high-speed impact experiments can be obtained by using the one-dimensional stress wave theory. The calculation formulas are (1), (2) and (3), as shown in section 2.2. The true stress-true strain curves that were obtained through SPHB tests are illustrated in figure 2. It can be seen from figure 2 that the stress values increased rapidly at the initial stage, and then with the increase of the strain, the flow stress gradually varied to the steady-state flow stage. The values of flow stress increase with the strain rates show the material has a positive strain rate sensitivity at varied temperatures. In the initial stages of dynamic deformation, strain hardening and strain rate hardening have an effect on the flow behaviors of 17 vol% SiCp / 7055Al composites. The true strain of the material are obviously improved under high strain rates, the higher the strain rate is, the larger the true strain. However, the true strain values are very low when the strain rate is around 400 s −1 , and there are no obvious plastic deformation stage. This was primarily because the amount of deformation was small, and the heat accumulation of the materials was not inadequate to achieve the significant increase in the dislocation motion.

Johnson-Cook model
The current JC model, considered the strain, strain rate and temperature effects on the plastic deformation mechanism of the metals. For its simple form and ease of use, the variables used have been already available in most calculation programs, and the model has been widely used in general the high-speed impact dynamics studies. The JC model is expressed as follow [25]: where s is the flow stress, A, B n, C and m are material constants, A is quasi-static yield stress (MPa) at reference temperature and reference strain rate, B is strain hardening parameter (MPa), n is strain hardening exponent, C represents the coefficient of strain rate hardening; m is thermal softening exponent. e p is equivalent plastic strain, and  e* is dimensionless equivalent plastic strain rate, which is expressed as /    e e e = .
Where, T is deformation temperature, T m is melting temperature of the composites at normal conditions and T r is the reference temperature. T r can be room temperature, the lowest temperature of interest or the lowest temperature of the experiment, and T m * cannot be negative. In the present experiment, the reference temperature is T r =293 K and the reference strain rate is  e 0 =0.001 s −1 to evaluate the material constants of the JC model. The elastic modulus and yield stress (A) can be obtained by quasi-static test, as shown in figure 3. It is found that the SiCp/7055Al composites have no obvious yield point, therefore, σ 0.2 was taken as the yield point, and the yield stress of the material is 242.29 MPa. The melting point of SiCp/7055Al composite is T m =900 K, the DSC curve as shown in  Then equation (6) can be denoted as: The parameters B and n could be obtained from the straight line fitted to the plastic deformation (after the yield point) of the quasi-static experimental data. The ln(σ-A)-lnε is plotted, and subsequently a linear fitting was performed, as shown in figure 5(a). The values of B and n could be calculated from the intercept and the slope of the fitting line, B =5383 MPa and n=1.33, respectively.

Determination of constant C
In equation (4), the second bracket on the right side of the equal sign indicates the strain rate enhancement effect, and the parameter C is the material strain rate sensitivity coefficient. At the test temperature of T=Tr=293 K, the relationship between the dynamic yield stress and the strain rate at normal temperature can be obtained as:  deformation, the JC model cannot accurately predict and loses its application value. It is therefore revealed that the original JC model cannot well describe the flow behaviors of the composite in the range of high strain rates, temperatures and strains. As a result, the JC model cannot comprehensively reflect the deformation mechanical properties at high-speed of the composites, to obtain a better prediction, the JC model needs to be modified.

Modified Johnson-Cook model
The JC model refers to a strain rate dependent constitutive model, considering strain, strain rate and temperature separately. These parameters can be determined by a few experiments. However, a considerable number of theories and experiments reported that shear modulus is a function of pressure and temperature [16,[26][27][28]. Thus, if the model is not modified, it cannot appropriately describe the stress-strain relationship of spray-deposition SiCp/7055Al composites at high-speed impact, as shown in figure 7. In addition to the effects of strain, strain rate, and temperature on the flow stress, the phase transition, dislocation density and material structure during deformation also have new effects on flow stress [16,[29][30][31]. To remedy the defects of the JC model, the coupling effects of the flow stress of these three factors were considered. Some modified JC models were proposed in the literature [22,23,[32][33][34]. These modified JC models can well describe the stress-strain   characteristics of theirs materials. The expression of the most widely used in these provided modified JC models is shown in equation (13). To determine whether this modified JC model is suitable for spray-deposition SiCp/ 7055Al composites, the equation (13) is verified. and  e ln * plots as shown in figure 9, in which the mean average slopes of the regression lines is the value of C 1 =0. 23.
Rearrange equation (13) and take natural logarithm on both sides, could be obtained equation (17), expressed as follow: The reason for the excessive errors is that the SiCp/7055Al composites not only has the strain rate hardening effect but also the softening effect, which is not taken into account in equation (18) [ 16,23,[35][36][37]. For some materials, the combined effect of thermal softening and strain hardening on the flow stress should be considered during dynamic deformation [22,23,38]. In view of the influence of thermal softening effect, it is reasonable to retain the thermal softening exponent in the temperature softening item. Considering the interaction influences of the factors on the flow stress, a modified JC model was proposed, as shown in equation (19). Rearranging equation (

Analysis of constitutive equation accuracy
To verify the reliability and practicability of modified Johnson-Cook model for the spray-deposition SiCp/ 7055Al composites at high strain rates, the experimental stress-strain values are compared with the stress-strain values predicted by the modified JC model, as given in figure 13. It can be found by comparing figures 7 and 13 that the proposed modified JC model exhibited accurate predictions with experimental results at different strain rates and varying temperatures. Since the modified JC model takes into account the coupling effects of temperatures, strain rate and strain rate softening effects on flow stress the accuracy of the proposed modified JC model are significantly improved. Note that in the initial deformation stage, the predicted values and the experimental values almost coincide at most conditions. However, when the strain is larger than 0.06, the predicted values gradually deviates from the experimental values, except at 523 k/971 s −1 . This is primarily due to the limited elongation of the material under the reference conditions. Thus, the prediction results are inaccurate once the impact deformation exceeds the material elongation. For the temperature at 523 K and the strain rate of 971 s −1 , the predicted values are more accurate when the strain is larger than 0.06, which may be due to experimental errors or changes in the microstructure of the material under this condition. Given this, the appropriate strain of the modified JC model should not exceed 10%. In fact, the addition of SiC particles greatly reduces the plasticity of the composites and improves the strength, so it is sufficient to predict the values of stress-strain in the initial stages of plastic deformation.
To compare the prediction accuracy of the JC model and the modified JC model, an error analysis was performed, as shown in table 3. The error between the predicted values and the experimental values was calculated by equation (22). Table 3, clearly shows that the commonly used modified JC model (equation (13)) and the original JC model errors were noticeably larger than those of the proposed modified JC model in this paper. The standard deviations of the JC model were 10.97% and 9.06% at 523 k and 573 k, while the standard deviations of the proposed modified JC model were 4.68% and 2.23% at 523 k and 573 k. However, the standard deviations of the modified JC-1 model at 523 K and 573 K are 15.75% and 34.43%, respectively. The modified JC-1 is not suitable for spray-deposition SiCp/7055Al composites at all. The proposed model can make accurate predictions the flow stress of the composites. The proposed modified JC model was built, and its effectiveness was verified based on the experimental data of spray-deposition 17 vol% SiCp/7055Al composites at high-speed impact temperatures range of 293 K-623 K and in strain rates range of 392 s −1 -2002 s −1 . The proposed modified JC model models can effectively assess the stress of the composite in the mentioned ranges.