Inuence of Size Effect on Dynamic Mechanical Properties of OFHC Copper at Micro/Mesoscopic Scale

The dynamic mechanical properties of metallic materials have been extensively investigated at the macro-scale in terms of deformation mechanisms, strain rate strengthening, and fracture mechanisms. However, the dynamic mechanical properties affected by size effects at micro/meso-scales have rarely been investigated. To explore the size effects on the dynamic mechanical properties at micro/meso-scales, the experiments of quasi-static compression and SHPB were carried out using oxygen-free, high-conductivity (OFHC) copper with different geometrical and grain sizes. The experimental results show that the quasi-static and dynamic mechanical properties of OFHC copper are affected by size effects at micro/meso-scales. In particular, OFHC copper exhibits strain rate strengthening effects at the micro/meso-scales, and the presence of micro-cracks was observed in the SHPB experimental specimens. The J-C constitutive model based on the surface layer model is proposed and the analysis of the average relative error of the modified model and the original constitutive model is performed. Finite element analysis was carried out based on the modified J-C model and the original model, and the results show that the modified J-C model was in good agreement with the experimental results.

The influence of size effects on the strength and formability of metallic alloy components was reviewed [4,5]. Jiang et al. [6] conducted dynamic shear tests using the hat-shaped samples with different shear ring thicknesses and the results showed that the shear stress and fracture strain increased as the shear ring thickness decreased.
Zhang et al. [7] found that the mechanical properties of the 7075-T6 aluminum alloy were influenced by the strain rate and showed a significant strain rate hardening effect.
Tanner et al. [8] have studied strain rate and temperature history effects on Oxygen-free high-conductivity (OFHC) copper. Work done by Smerd et al. [9] shown that Al-Mg alloys AA5754 and AA5182 exhibited low strain rate sensitivity at both room and high temperatures. The dynamic mechanical properties of selective laser melted Ti-6Al-4V alloy exhibits an increase in yield stress with increasing strain rate, showing a significant strain rate sensitivity. This phenomenon can be explained as an increase in plasticity induced by the strain rate effect [10]. Mechanical experiments on different metal and alloy materials at high strain rates were conduted to study the mechanical properties of the materials under dynamic loading and the deformation mechanism [11][12][13][14].
The constitutive model is usually used to describe the functional relationship among strain, strain rate, and temperature in plastic deformation of materials.The constitutive model of metal under a high strain rate can be divided into two kinds [15]. One is the 4 empirical constitutive model, including Johnson-Cook (J-C) constitutive model [16], Khan-Huang (KH) model [17], and some other empirical models [18][19][20]. The other one is the physically based constitutive model, including Zerilli-Armstrong (Z-A) constitutive model [21], Mechanical Threshold Stress (MTS) constitutive model [22], and NN-Li constitutive model [23]. The characteristic of empirical equations is that stress can be expressed as functions of strain, strain rate, and temperature and their parameters can be easily obtained from experimental data. Physically based constitutive model can reflect the micro-scale changes of material during the deformation process such as dislocation, thermal activation, and recrystallization. Huh et al. [24]  H. K. et al. [25] determined the parametersof the J-C model for Inconel 718 by quasi-static tensile tests, flat plate impinging on a steel ball and simulation using ABAQUS. Ducobu et al. [26] gathered twenty sets of J-C parameters for Ti-6Al-4V and put into a Coupled Eulerian-Lagrangian (CEL) finite element orthogonal cutting model. Then an accurate method for cutting were obtained by comparing all numerical simulations with experimental results in the same condition. However, the original model is not universal. J-C model was modified in many researches. Zhang et al. [7] modified the strain rate hardening term of the J-C model through experimental data, which was more accurate than the original J-C model. Gambirasio et al. [27] introduced a modified J-C model, namely Split Johnson-Cook model, and this model solved the problem that the equivalent plastic strain rate and temperature have different effects on each equivalent plastic strain. However, the dynamic mechanical 5 properties of materials at micro/mesoscopic scales have rarely been studied. Moreover, the original and the modified J-C models are mostly applied to the prediction of the dynamic properties of materials at macro-scale, and the size effects are rarely taken into account in the model for modification and application to the dynamic mechanical properties at micro/meso-scales.
In this study, OFHC copper of different geometrical sizes with different grain sizes were used to investigate the influence of size effects on dynamic mechanical properties. The specimens were subjected to quasi-static and split Hopkinson press bar tests at micro/meso-scales. The metallographic and microstructure of the specimens after the experiments were observed to investigate the deformation mechanism at the micro/meso-scales. A modified J-C model based on the surface layer model was proposed to study the size effects. The parameters were fitted using experimental data from quasi-static and separated Hopkinson pressbar tests, and the proposed model was verified through the experiments.

Workpiece materials
The commercially oxygen-free high-conductivity (OFHC) copper was chosen as the experimental material, which was widely used in the electronic and optical components. The shape of the specimen is cylindrical with a height to diameter ratio of 1:1. The diameter of the specimen is 0.5, 1, and 2 mm. The 1mm specimens were annealed at 450 °C for 2h, 600 °C for 2h, and 750 °C for 3h to obtain different grain sizes and to study the effect of grain size on the mechanical properties of OFHC copper at the micro/meso-scales. Heat treatment conditions as shown in Table 1. The heat-treated specimens were grinded, polished, and etched to obtain the metallographic microstructure, as shown in Fig. 1.

Experimental setup
The quasi-static mechanical properties of the specimens were obtained by compression tests on a universal testing machine with a pressure transducer of 5 kN.
In order to ensure the accuracy of the experiment, a stamping fixture with low surface roughness values and high flatness were designed. Lubricants were also applied to the ends of the specimens to further reduce the effects of friction. Each set of experiments was performed three times to eliminate experimental chance errors. Firstly, the quasi-static compression tests were carried out on specimens of different grain sizes to 7 investigate the grain size effect on the quasi-static mechanical properties of OFHC copper. Quasi-static compression tests were then carried out on different feature sizes to investigate the geometrical size effect on the quasi-static mechanical properties of OFHC copper. Compression tests were also carried out on 1mm specimens at 30%, 40%, 50% and 70% downward displacement to investigate the plastic deformation process of the specimens, and additional compression tests at 50% downward displacement were carried out on 0.5mm and 2mm to investigate the effect of size effect on the mechanical properties.
The dynamic mechanical properties of the specimen were obtained by using a split Hopkinson press bar device (SHPB). The impact bar of tests was blown through the gas in the storage chamber, and the air pressure is vaired in the chamber for different strain rate experiments. The ends of the specimen are applied grease to reduce the effects of friction. The analysis of the one-dimensional stress waves generated by the incident and projected rods utilizing a hyper-dynamic data acquisition system to obtain the dynamic stress-strain curves of the specimen. SHPB tests were performed at strain rates of 7000, 9000, and 13000 s -1 to investigate the grain size effect on the dynamic mechanical properties of OFHC copper. The 0.5 and 2mm specimens were also annealed at a temperature of 600 °C for 2h, followed by SHPB tests to investigate the grometrical size effect on the mechanical properties. with the increase of grain size. This phenomenon has been associated with peer researches [3,28]. The reason for this is that when the geometrical size is the same, the increasing in grain size increases the ratio of surface grains to all grains, which 8 makes it easier to torsion and deformation as the surface grains are less constrained.The original specimen without heat treatment was influenced by the previous machining process, and had the highest flow stress. Due to frictional effects the true stress-strain curves of the 600 °C annealed specimens are very close to those of the 750 °C annealed specimens. From Fig.3, it can be seen that the reduction in flow stress of the material decreases with decreasing geometric size for the same grain size, which is consistent with Chan's research [29], and shows the phenomenon: "the smaller, the weaker". When the grain size is the same, the smaller geometrical size leads to an increase in the number of surface grains. In the plastic deformation process, the surface grain is more prone to plastic deformation and the flow stress of the material is reduced.

SHPB experiments
As can be seen in Fig. 4, for the same geometrical and grain size, the flow stress of the specimen increase with increasing of strain rate, which shows a significant strain rate strengthening effect; and the degree of plastic deformation of the material increase with increasing of strain rate, which shows a strain rate induced plasticisation . This is inconsistent with the findings of most existing studies of macroscopic OFHC copper under dynamic loading, and there is uncertainty in strain rate sensitivity at macro-scale [24,30,31]. Zoller et al. [32] found that when the geometrical size was reduced, an increase in strain gradient led to an increase in dislocation density. According to Orowan's relationship, strain rate and stress are influenced by dislocation velocity and dislocation density [33,34]. The strength of the material increases with increasing of dislocation density over a range of dislocation densities. When the sizes of the specimen reduce to micro/meso-scales, the strain gradient during the deformation under external dynamic loading lead to increasing of dislocation density. The increasing of dislocation density affects the dynamic mechanical properties of the specimen, which is why OFHC copper exhibits strain-rate strengthening effects at the micro/meso-scales. It can also be seen through the grains, which leads to the fluctuations of the true stress.In Fig. 4 (b), the true stress-strain curve with a strain rate of 9000s -1 is significantly higher in the elastic stage and the initial stage of plastic deformation than the true stress-strain curve with a strain rate of 13000s -1 , possibly due to frictional effects and the instability of the air pressure during the experiments. As can be seen from Fig. 5, the flow stress of the specimens with the same grain size decreases as the decreasing of geometrical size under the same strain rate, which proves the phenomenon of "the smaller, the weaker".
There was a slow growing stage in the true stress-strain curve of dynamic compression . In the dynamic compression process, the dynamic load response time is short and the energy generated by plastic deformation can not be released, which leads to the increasing of temperature in the specimen and the emergence of thermal softening effect. So the flow stress appears to have a slow growing stage.

Metallographic and micro-morphological observations
The specimens were collected after the experiments, and then polished and etched to obtain metallographic photos, as shown in Fig. 6. The internal grains are compressed and elongated in the direction perpendicular to the direction of compression. After quasi-static and dynamic compression, the grains underwent elastic deformation and severe plastic deformation, the internal grains were compressed and elongated as shown in Fig. 6. Due to the short dynamic compression time and high impact velocity, the grains of the dynamically compressed specimens are flatter and more elongated than those of the quasi-statically compressed specimens, and even some grain boundaries are almost compressed together. Severe plastic deformation resulted in micro-cracks on the surface of the specimen in Fig.6 (a), which are not observed in the quasi-static compression specimen in Fig. 6 (b). The specimens after the quasi-static and dynamic compression experiments take the shape of a bulgen caused by friction, as shown by the dashed line in the Fig. 6. The flow direction of the circumferential surface grains is oriented at 45° to the direction of compression, as shown by the arrow in Fig. 6. At the micro/meso-scales, dynamic loading and random grain distribution exacerbate the variation in grain orientation and inhomogeneous properties.   Using the above equation, the adiabatic temperature rise can be calculated for 1mm specimens with different grain sizes and at different strain rates, as shown in Fig.10. It can be seen from Fig.10 that the adiabatic temperature rise is increasingly obvious with the increasing of strain rate for the same grain size. The adiabatic temperature rise with the increasing of grain size for the same strain rate. The proportion of surface grains increases with the increasing of grain size, and the resistance of surface grains to deformation is lower than interior grains , then lower plastic work is converted to heat energy during deformation.

Surface layer model
As the geometrical size of the specimen decreases, the ratio of surface grains to all grains increases. It is known from crystal plasticity mechanics that the free surface cannot transfer and store dislocations, and the surface grains are less constrained and more likely to twist and deform, which causes the flow stress to decrease as the size of specimen decreases during plastic deformation. The surface layer model considers the flow stress of the material to consist of the flow stress of surface grains and the internal grains, and the flow stress of the material can be written as: Where  is the total flow stress, N is the total number of grains, s N is the total number of surface grains,  is the ratio of surface grains to total grains, which is called the size factor, s  is the flow stress of superficial grains and i  is the flow stress of internal grains.
Base on the surface layer model, Lai et al. [35] proposed a hybrid intrinsic structure model that takes into account size effect. In this model, the surface grains are considered as a single crystal and the internal grains are considered as polycrystals.
According to crystal plasticity theory and the Hall-Petch equation, the flow stresses in the surface and internal grains can be written as: where d is the grain size, m and M are the orientation factors for single and polycrystalline grains respectively, () R  is the principal decomposition shear stress of individual grains, and ( ) k  is the resistance stress at the grain boundary.
Combining Eq. (2) and Eq. (3), the flow stress of the material at the micro/meso-scales can be written as: When the ratio of surface grains to total grains is equal to zero, i.e.  =0, it means that the geometry of the material is much larger than the grain size, and ( )  represents a polycrystalline material model, when the ratio of surface grains to total grains is equal to one, i.e.  =1, it means that the geometry of the material is much equal to the grain size, and ( )  represents a single crystal material model. Further simplification of Eq. (4) was given in [1]: Where dep  is size-dependent flow stress and ind  is size-independent flow stress.
The specimens used in this study are cylinder and the relationship between the surface grains and the internal grains at the microscale is shown in Fig. 11. The diameter of the cylinder is D and the grain size is d. The volumes of the surface grains and the internal grains are as follows: Fig.11 The surface layer model of the micro/meso-scales cylindrical sample. 20

The modified J-C model
The J-C model considers the flow stresses of metallic materials influenced by strain, strain rate, and temperature at deformation, and is widely used in the study of dynamic mechanics because of its simple form and easily available parameters. It can be written as: Where A is the initial yield stress of the material; B is the strain hardening coefficient; n is the hardening index; C is the strain rate strengthening coefficient; m is the thermal softening index; all of the above parameters can be obtained experimentally;  is the equivalent plastic strain; 0  is the reference strain rate;  is the equivalent plastic strain rate; T is the actual temperature; r T is the room temperature; m T is the melting temperature.
According to the above study, the quasi-static and dynamic mechanical properties of the material are affected by size effects at the meso/micro-scales, so the size effects should be considered in the modified intrinsic structure model. Wang et al. [1] found that a hybrid damage model consisting of a surface layer model and the Hall-Petch equation was in good agreement with experimental results in predicting the quasi-static mechanical properties of materials at the micro/meso-scales. Therefore, the first term of the J-C model is replaced by a hybrid model. In this study, the strain-rate strengthening term is introduced to correct the term of strain rate in a similar form, taking into account the effect of size effects on dynamic mechanical properties. The modified J-C model can be written as:   R  and   k  can be viewed as exponential functions related to plastic strain and can be written as: In this study, the specimen with 1 mm was chosen to fit for the parameters of the constitutive equation, as several grain sizes and feature sizes were involved. In particular, the change in   k  was found to converge to a constant function in the parameter fitting, so   k  was set as a constant function. The fitted parameters of the original J-C and modified J-C are shown in Table 2 and Table 3.   Using the average relative error  to analyze the experimental curve and fitted curve data, the relative error formula can be written as: Where i E is the experimental value, P i is the fit value of the model, and Z is the total amount of data. The

Numerical simulation
In this study, in order to verify the accuracy of the modified model, simulations were

Conclusion
In this study, the microstructure of OFHC copper with different grain sizes was obtained by heat treatment, and the influence of size effect on the quasi-static and dynamic mechanical properties of OFHC copper at the micro/meso-scales was

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.