Microscopic defects formation and dynamic mechanical response analysis of Q345 steel plate subjected to explosive load

Zhengqing Zhou , Zehen Du , Yulong Zhang , Guili Yang , Ruixiang Wang ,Yuzhe Liu , Peize Zhang , Yaxin Zhang , Xiao Wang

a Research Institute of Macro-Safety Science, University of Science and Technology, Beijing,100083, China

b School of Civil and Resource Engineering, University of Science and Technology, Beijing,100083, China

c China North Industry Advanced Technology Generalization Institute, Beijing,100089, China

d College of Field Engineering, PLA Army Engineering University, Nanjing, 210007, China

e School of Mechanical Engineering, Nantong University, Nantong, 226019, China

Keywords:Explosive load Q345 steel Micro defect Finite element simulation Dynamic response Data fitting

ABSTRACT As the basic protective element, steel plate had attracted world-wide attention because of frequent threats of explosive loads.This paper reports the relationships between microscopic defects of Q345 steel plate under the explosive load and its macroscopic dynamics simulation.Firstly,the defect characteristics of the steel plate were investigated by stereoscopic microscope (SM) and scanning electron microscope(SEM).At the macroscopic level, the defect was the formation of cave which was concentrated in the range of 0-3.0 cm from the explosion center, while at the microscopic level, the cavity and void formation were the typical damage characteristics.It also explains that the difference in defect morphology at different positions was the combining results of high temperature and high pressure.Secondly, the variation rules of mechanical properties of steel plate under explosive load were studied.The Arbitrary Lagrange-Euler (ALE) algorithm and multi-material fluid-structure coupling method were used to simulate the explosion process of steel plate.The accuracy of the method was verified by comparing the deformation of the simulation results with the experimental results, the pressure and stress at different positions on the surface of the steel plate were obtained.The simulation results indicated that the critical pressure causing the plate defects may be approximately 2.01 GPa.On this basis, it was found that the variation rules of surface pressure and microscopic defect area of the Q345 steel plate were strikingly similar, and the corresponding mathematical relationship between them was established.Compared with Monomolecular growth fitting models (MGFM) and Logistic fitting models (LFM), the relationship can be better expressed by cubic polynomial fitting model (CPFM).This paper illustrated that the explosive defect characteristics of metal plate at the microscopic level can be explored by analyzing its macroscopic dynamic mechanical response.

1.Introduction

As the basic protective structural component, steel plate is widely employed on bridges, construction industry, nuclear industry, offshore platforms, and shipbuilding industry [1-3].However, these carbon structural steel components utilized both in civilian and military engineering are frequently subjected to the explosive loads [4-6], which will inevitably lead to the material deterioration and life reduction.The sudden release of energy from an explosion could give rise to an instantaneous high-pressure wave [7,8], coupled with the violent expansion of hot gases, moving outward at high velocity from its source.Thus, the dynamic loads originating by explosions are impulsive, and could result in the extreme damage to the structural steel components[9-11].

From macroscopic level, many explosion experiments and modern numerical simulation technology were being used to investigate and predict the macroscopic explosive deformation of the metal plates and panels [12-16].Rudrapatna et al.[17]conducted the explosive impact simulation upon the stiffened steel plates that based on the super element method, and then made a comparison with the experimental displacement data, and developed a node release algorithm to simulate the progression of rupture;Micallef et al.[18]presented an analytical formulation for the permanent plastic deformation of a circular plate subjected to an explosion load.A series of numerical simulations were obtained by using ABAQUS and excellent correlation with the derived analytical results (within 15% accuracy) are obtained; Zheng et al.[19] investigated the structural response of mild steel plates by using the multi-material Euler formulation, and made the reliability verification on deflection between the experimental and numerical results.They found that the maximum plastic deformation decreases greatly with increasing the mass ratio of the stiffener to the steel plate.Shishegaran et al.evaluated the resistance of steel plate shear walls(SPSW)[20]and reinforced concrete panels (RCPs) [21] under explosive loads using nonlinear finite element analysis and surrogate methods.The effect of several design parameters on the maximum deflection and damage of the two under explosive loads has been evaluated.The results showed that the cross-section in the y-direction and the plate thickness have the most significant effects on the maximum deflection of SPSWs and the explosive weight and distance from the explosive have the most impact on the RCP failure,respectively.

From microscopic level, microstructure of steel determines the corresponding mechanical properties and electron microscope is an effective method for diagnosing and observing the microstructural change before and after loadings[22-24].Rajendran et al.[25]observed the characteristics at the fracture location of HSLA steel plates.The Scanning electron microscopic (SEM) examination of dimples at the fractured surfaces suggested that crack propagation was caused by the coalescence of microscopic voids.Geffroy et al.[26] observed the fracture morphology of a high purity ferriticpearlitic DH36 steel plate under the spherical explosion load, the results showed that the fracture is characterized by shear ductile fracture.

In conclusion,many efforts have been performed to investigate the material damage of steel plate subjected to the explosive loads.Most studies such as the mentioned above are mainly focused on the exploration of the macroscopic dynamic mechanical response and the microscopic damage mechanism of the structural steel components, but are poor at combining the two to explore the detailed information of corresponding relationship between them.Until now,the research and understanding of the explosive damage of structural steel are still far from enough,further research should be devoted to this area.

In this study, typical structural steel, i.e., type Q345 steel was selected as the research object, the explosive damage feature and dynamic mechanical response of this material were investigated at the microscopic level by using various characterization methods and dynamics simulation in combination.What's more,the mathematical relation between the two was established.The results can not only reveal the relationship between macroscopic dynamic mechanical response and microscopic defect of explosive damage of steel components, but also provide theoretical basis, data support, and method guidance for predicting and preventing damage propagation and improving the protection ability of materials.

2.Experiments and numerical simulation

2.1.Specimen preparations and explosion experiments

The normal Q345 steel plate, with following chemical composition(see Table 1),was used in this study.As shown in Fig.1(a),the dimension of this steel plate was 200 mm×200 mm×20 mm (i.e.,length×width×thickness),the weight of cylindrical RDX(RDX 95%and wax 5%) was about 300 g, and its diameter and height were 54 mm and 52 mm,respectively.The RDX was placed 20 mm above the center of the steel plate, and was detonated by using the detonator with JH-14 booster.The experiment was carried out in an explosion chamber, the diameter and height of it were 7 m (see Fig.1(b)).After the explosion,a rounded dent with radiated traces was formed on the plate surface,which can be seen in Fig.1(c).To measure the deformation extent, the steel plate was cut along the dashed line (see Fig.1(d)) by using the wire electrical discharge machining.Then, the displacement of this plate was measured by using the graduated scale and the maximum displacement after the explosion was about 16.5 mm.

Table 1 The chemical composition of this Q345 steel.

2.2.Microstructure observation

To investigate the damage of Q345 steel at different positions from the explosion center,several samples were prepared from this plate for the microstructure observation.Samples were collected by using the wire electrical discharge machining.First, the steel plate was cut along the dashed line (see Fig.2(a)) to get one quarter of this plate.Then, the obtained small plate was cut along the white line (see Fig.2(b)) to seven small samples.Here, the dimension of each sample was 12 mm×12 mm×20 mm, and were named from 0# to 6# respectively.

The surface microscopic defect was preliminarily observed for each sample by using Zoom-stereo microscope (ZS7045) and its deeper observations were made using SEM (SUPRA55).The defect size fraction was counted by using the image analysis software(Image-Pro Plus 6.0).

2.3.Model building and numerical simulation

The finite element(FE)model was built by using the commercial code LS-DYNA which can deal with dynamic response problems with large deflection and high strain.Arbitrary Lagrange-Euler(ALE) algorithm provided by LS-DYNA was selected in this research to realize the dynamic analysis of fluid-structure coupling.A quarter 3D model was created to simplify the calculation and improve the accuracy.

In the simulation,the Jones-Wilkins-Lee(JWL)Equation of State(EOS)[27]was implemented to describe the RDX,which is defined a

where P is the product pressure,e is the energy per unit volume,υ is the relative volume of the gas products to the initial explosive state,A, B, R1, R2,ω are constants.

To obtain the JWL-EOS parameters, cylinder tests was performed (see Fig.3(a)).According to the result of the wall radial velocity in cylinder tests, the JWL-EOS parameters were determined by iterating these variables in LS-DYNA hydrocode simulations until the experimental values were reproduced(see Fig.3(b)).The obtained JWL-EOS parameters[28] were shown in Table 2.

In addition, the medium in which the blast wave propagates(air) was modeled with a linear polynomial EOS for linear internal energy, which is given by

Fig.1.Schematic diagram of this explosion experiment:(a)The Q345 steel plate and RDX;(b)The explosion chamber;(c)Macroscopic examination of the Q345 steel plate before and after explosion experiment; (d) Deformation of Q345 steel plate and its displacement curve.

The Johnson-Cook strength model was used to describe the dynamic behavior of metal materials.The Johnson-Cook constitutive equation [30] was given as follows:

where C0-C6are the constants related to the material properties,C1=C2=C3=C6=0.μ=V-1,V is the specific volume,V=ρ/ρ0,ρ is the density, ρ0is the initial density; E is energy density, namely energy per unit volume.The parameters [29] were shown in Table 3.

where Y is the dynamic yield stress of the material, ε is the equivalent plastic strain, έ is the equivalent plastic strain rate, έxis the reference strain rate,A1,B1,n,C and m are material parameters,T0is the reference temperature and Tmis the melting temperature.

Fig.2.Schematic diagram of Q345 steel plate cutting and sample preparation:(a)The Q345 steel plate was cut to the size of a quarter;(b)The cut Q345 steel plate was divided into seven identical samples (named 0#-6#).

Fig.3.Acquisition of experimental parameters: (a) Expansion diagram of copper tube in cylinder test; (b) The comparison between simulation and experimental results.

Table 2 The JWL-EOS parameters of RDX.

Table 3 The Linear polynomial EOS coefficients of air.

The Q345 steel has strict production standards in China,i.e GB/T 1591-2008[31].The parameters of Q345 steel [32]were obtained by universal material testing machine, Hopkinson press bar experiment (SHPB) and Gleeble thermal simulation testingmachine[33].The values of material parameters of Q345 steel plate were shown in Table 4.

Table 4 Johnson-Cook parameters of Q345 steel.

3.Results

3.1.Macrostructures observation by stereoscopic microscope (SM)

The SM was used to observe the surface macroscopic defects of Q345 steel plate firstly.Each sample was tested three times, since the observation results of the same sample were consistent,Figures that could better show the defect characteristics were selected for follow-up study.Fig.4(f) showed the original surface morphology of the Q345 steel.It can be seen that there was no obvious damage appeared.The surface morphology of the 0#sample was shown in Fig.4(a).It can be observed that, within 12 mm of the explosion center, many large caves were generated under explosive load.The dimension of these caves was nearly 0.5-2 mm, and they have reached a deep degree of depth.The geometry configuration of them was irregular, including ovalshaped, strip-shaped, and peanut-shaped, there were upward bulges at the walls of the caves, and the bulges were smooth without edges and corners.Meanwhile,some of rusty spots can be observed inside the caves, which could be formed during the sampling process with water flow.Besides, at relatively higher magnification, many white particles were observed around and inside the caves,those particles could be the iron oxide that formed at high temperatures, which was markedly different from those rusty spots.

The surface morphology of the 1# sample was shown in Fig.4(b).In the range of 12-24 mm from the explosion center,the cave generation was still the main feature of explosive damage.However,the size and depth of those caves seem to be smaller than that of the 0#sample,which indicated that the damage reduced as the distance from the explosion center increased.This can be further recognized by observing the surface morphology of the 2#sample, as shown in Fig.4(c), in the range of 24-36 mm from the explosion center, a clear transition region can be seen on the sample surface, some small caves can still be observed at the leftside area, while the right-side area was flat and without any cave in this place.Fig.4(d) showed the surface morphology of the 3#sample.In the range of 36-48 mm from the explosion center,there was no longer any cave in this area.A lot of stripe-like scratches can be only observed on the sample surface,which were left over from the machine work.From 3# to 6#samples,no obvious differences can be found on the surface morphology by using the SM observation, so the micrograph of 4# and 5# samples were no longer presented here.

In conclusion, under the impact of explosive load, the macroscopic damage type of steel plate surface is relatively single (only cave),and concentrated in the range of 30 mm from the explosion center, the reasons for this damage are as follows:

Fig.4.SM Figures of the samples: (a)-(e) Macrostructure morphology of Q345 steel sample under impact load; (f) Macrostructure morphology of the original Q345 steel.

The radius of the cylindrical explosive is 27 mm.When the cylindrical explosive was detonated on its end face,the energy on the cylindrical surface diffused outward to some extent,and the shock wave pressure along the axis of initiation was higher, which had obvious directional gain.Therefore,the damage was most serious in the area directly below the bottom surface of the explosive,and the caves were concentrated in this range.In addition, due to the uneven explosive charge,the shock wave generated by the explosion did not propagate outward along the ideal continuous and uniform wave front, which can be regarded as a dispersed regional impact on the surface of the target plate.Because of the coupling effect of high temperature and high pressure,the steel on the surface of the target plate melted, many caves of different sizes and irregular shapes were generated,and smooth accumulation appeared at the edges of caves.

3.2.Microstructures observation by scanning electron microscope(SEM)

The SEM was used to observe the surface microscopic defects of Q345 steel plate.Two magnifications were presented here for the investigation of defect characteristics.Fig.5(a) showed the SEM images of the 0# sample, at low magnification (i.e., 30 × ), it seemed that the morphology,contour,or dimension of these caves cannot be distinguished very well as compared with the observation results that were shown in Fig.4(a).However, the detailed feature of the defects was revealed well by SEM images at high magnification(i.e.,1000×).From 1000×image in Fig.5,it can be seen that there were two different types of microscopic defects generated after explosion.Each sample was tested twice under the same conditions, since two observation results were similar, the Figures which can better characterize the microscopic damage effect of the sample,were selected here for follow-up study.The area of defects was measured by the Image-Pro Plus 6.0,30 microscopic defects were randomly selected from each sample, and their characteristics (defect area and diameter) were counted.The obtained data were averaged,and the calculation results represented the defect characteristics of the samples.According to the statistics(see Table 5), starting from 3# sample, the maximum defect areas were almost less than 50 μm2.Therefore,we specified that defects larger than 50 μm2were named Type 1 cavity,and defects smaller than 50 μm2were named Type 2 void.

A lot of Type 1 cavities which had many small through-cavities were generated under the explosive load,most of them were ovalshaped.Meanwhile, there were also Type 2 voids exist around the cavities, which were marked in Fig.5(a).Besides, the Energy Dispersive Spectrum (EDS) was carried out for surface oxides of steel plate.The result showed that,on the surface of 0#sample,the content of oxygen element and iron element were the highest at the test point, which were 63.71% and 19.72% respectively (see Table 6).It was further confirmed that the oxide came from the oxide of iron after explosion.

The surface morphologies of the 1#and 2#samples were shown in Fig.5(b)and 5(c),respectively.It can be observed that the main feature of explosive defect was still the formation of Type 1 cavity.The area and diameter of cavity of the 2#sample were smaller than that of the 1# and 0# samples, the cavity number was also decreased, which meant the explosive defect of plate reduced as the distance to the explosive center increased.Nevertheless,though the number of Type 1 cavity gradually decreased in the first three samples,the number of Type 2 void increased significantly in the 2# sample as compared with the 0# and 1# samples.The surface morphologies of the 3# and 6# samples were shown in Fig.5(d)and 5(e),different from the 1#and 2#samples,there were no longer any Type 1 cavity in the rest samples.In contrast, the main defect feature of the 3#-6# samples was the formation of Type 2 void.

It is worth noting that the cavities and voids of the 0# and 1#samples (within the range of 24 mm from the explosion center)were large in size,regular in shape and smooth in surface,while the cavities and voids of the other samples were small and rough.This phenomenon may be caused by the following reasons:

Fig.5.SEM Fig.s of the observation points: (a)-(e) Microstructure morphology of Q345 steel sample under impact load;(f)Energy spectrum analysis(EDS)of Q345 steel sample after explosion.

Table 5 The area of defect of samples.

Table 6 The EDS results of the test point.

Under extreme loads such as explosion, high speed and ultrahigh speed collision, the process of defects was due to the nucleation and growth of micro voids,which were connected into micro cracks,and the confluence of micro voids and micro cracks to form large voids and cavities [33-35].0# and 1# samples were the closest to the explosion center,where the pressure was the largest,so the micro voids were the first to form here and merge with each other to form the larger cavities.Finally, the macroscopic caves in Fig.4 were formed.Starting with the 2# sample, some molten metal oxides can be observed on the samples’ surface.The formation of those might be described as follows: the instantaneous release of high-temperature and high-pressure wave during the explosion could lead to the melting of the metal plate surface at its central position.The molten alloy droplets first splashed from the explosive center to the surrounding, and then solidified and deposited on the plate surface, which appeared as the flake and spherical deposition.Therefore, the surface morphology of the samples near the explosion center was the smoothest, while the surface of other samples was relatively rough.

In conclusion, it can be preliminarily obtained that with the increase of the distance from the explosion center,the pressure and temperature on the surface of the steel plate gradually decreased,and molten material and the cavity correspondingly decreased.

Fig.6.The overall trend of the area and diameter of defects and its fitting model: (a), (c) The PFM of defect area and diameter; (b), (d) The EFM of defect area and diameter.

The average value of the characteristic data of microscopic defects showed that the defect area and diameter decreased with the increase in the distance from the explosion center.To describe the damage rule of steel plate more accurately,the area and diameter of defects were fitted by the cubic polynomial fitting model (CPFM)and the exponential fitting model (EFM), respectively (see Fig.6).With respect to the defect area, the performance of CPFM was satisfactory within 2.5 cm from the center of the explosion, however, the deviation error was relatively large outside this range;In contrast, the fitting result of EFM at the range of 2.5-7.5 cm was better than the former.Therefore,a piece-wise function model was constructed by combining both the CPFM and the EFM(see Eq.(4)),and this model can describe the change of defect area well; With respect to the defect diameter,the fitting results of the two models were similar within the first 2.5 cm.Outside this range, the deviation of the CPFM was large,which could not represent the change rule of defect diameter.On the contrary,the EFM can fit the defect diameter quite well, which can be used as the target model to estimate the variation of defect diameter(see Eq.(6)).

3.3.Numerical simulation of dynamic mechanical response of Q345 steel plate

In order to improve the computational efficiency, a quarter model was established and the displacements in all directions of the edges of the steel plate were fixed, as shown in Fig.7(a).All components in the model have meshed with 3D 8-node element SOLID164 as indicated in Fig.7(b).The ALE algorithm was used to simulate the explosion of Q345 steel plate, and the numerical simulation results were compared with the experimental results to verify its accuracy.

The Q345 steel plate was simulated and analyzed with five mesh sizes (i.e., 0.8,1.0, 4.0, 8.0, and 12.5 mm).Figs.8 and 9 show the results of experiment and simulation with various mesh sizes.In Table 7, compared with experimental result, the mesh size of 1.0 mm had the smallest error (i.e., 17.38%), and the maximum displacement at the center of the steel plate in the simulated model and experimental test were 16.4 mm and 16.5 mm, respectively.Therefore,the model with mesh size of 1 mm was finally selected.

Fig.10 displayed the contours of pressure, effective stress and effective plastic strain,the dynamic mechanical response process of steel plate during explosion can be recognized here.The pressure on plate rose sharply at first and then fell later.At 10 μs, the pressure on the plate was just 9.743×10-5GPa.And from 36 to 184 μs,this value decreased from 1.113 to 0.484 GPa,which can be seen in Fig.10(a);The effective stress presented the similar variation rule of pressure.At 10 μs, the effective stress of this plate was just 6.935×10-5GPa.And from 36 to 184 μs,this value decreased from 9.583 to 7.211 GPa, which can be seen in Fig.10(b); Different from pressure and effective stress, the effective plastic strain increased first with time, and then remained a constant value after that.As shown in Fig.10(c), it can be observed that, from 10 to 36 μs, the effective plastic strain of steel plate increased from 0 to 0.633,and then, it remained at 0.739 after 80 μs.

Fig.7.The boundary condition of the simulation and the considered elements for steel plate:(a)Boundary condition of steel plate;(b)The considered element for steel in LS-DYNA.

Fig.8.The simulation results of the simulation with various mesh sizes.

Fig.9.Comparison of experimental and numerical results:(a)The displacement of experimental and simulation result;(b),(c)The explosion crater of experiment and simulation.

Table 7 The comparison of various mesh sizes.

The variations of pressure and effective stress in such ways could be explained for the following reasons:Since the detonation reaction time of RDX was very short and the propagation speed of shock wave was fast, the pressure and effective stress rapidly increased on the plate surface in a relatively short period.After that,the shock wave decayed,the pressure on the plate surface gradually decreased, and so did the effective stress.In contrast, the effective plastic strain increased first under the shock wave pressure.But when the pressure decreased, it was not enough to cause the further deformation of the steel plate, so the strain remained unchanged after rising to the maximum value.Besides, it can be observed that the pressure, effective stress, and effective plastic strain were higher at explosion center, but lower away from the explosion center.

Nevertheless, the exact data of steel plate cannot be read directly in the above contours,so the specific values of each point at different distances from the explosion center (i.e.P0-P6) were extracted in the post-processing for investigating the variation laws of above parameters.As shown in time-history curves (see Fig.10(d)), the changes of those parameters were consistent with the above contours: the pressure and effective stress on plate increased remarkably at the first 25 μs and then fell in the followup, while the effective plastic strain increased at first and then remained unchanged; Meanwhile, all of the above parameters of the first two samples(i.e.0#and 1#)were significantly higher than the other samples (i.e.2#-6#).

In addition,more details can be observed in time-history curves:During the period of 80-120 μs,the effective stresses of each point changed obviously, especially for the P0and P1(see black arrows).The variation of effective stresses could be attributed to pressure change of the sample.The pressure on P0and P1also had a similar variation rule during about 80-120 μs (see blue arrow).The variation of pressure on these points could be due to the following reason: For P0, the pressure first passed down from the exploded surface (top) to the free surface (bottom), and the pressure amplitude of this point reduced at this process; after that, the pressure touched the free surface and reflected the exploded surface,which led to the pressure amplitude increased of this point.

Besides, from the effective plastic strain curve shown in Fig.10(d),it can be observed that at different positions on the plate,the closer to the explosion center, the greater the plastic deformation of the steel plate, and the longer the time to reach this deformation.As P5and P6were far away from the explosion center,there was almost no obvious deformation,on the contrary,point P0and P1were located directly below the RDX, thus the values of above parameters of P0and P1were much larger than those of other points.

4.Discussion

Under the impact of the explosion, metal materials and structures could exhibit complex failure mechanisms[36].Liu et al.[37]showed that: For plate structures, different local stress states caused by the explosive load could lead to different failure mechanisms,such as a tensile or shear ductile fracture.In this study,the damage to steel plate was closely related to its dynamic mechanical response.The microscopic morphology feature and numerical simulation results were obtained from the above works.The change rules of cavities at different positions from the explosion center as well as the variation laws of pressure,effective stress and effective plastic strain of steel plate were explored,respectively.

Generally, the cavity formed on the surface of metal materials caused by external force was related to the maximum stress or strain which it suffered [38-40].The maximum stress-strain state of the steel plate can be preliminarily recognized from the contours(see Fig.11(a)).Furthermore, the following statistical method was adopted to obtain the more specific and precise values of each sample (0#-6#).Since each sample was composed of 144 grids,and the values of a single grid cannot accurately represent the stress-strain state of sample.Therefore, multiple grids were selected to average the sample, as shown in Fig.11(b), in one sample,12 grids were selected from the outside to the inside, the average value of each parameter was taken, and the results were used to study the variations of pressure, effective stress, and effective plastic strain of samples at a different distance from explosion center that produced by the explosive load.

Fig.10.Dynamic mechanical response processes of Q345 steel plate model: (a) Contours of pressure of Q345 steel plate model (The units are 100 GPa); (b) Contours of effective stress of Q345 steel plate model(The units are 100 GPa);(c)Contours of effective plastic strain of Q345 steel plate model;(d)Time-history curves of pressure,effective stress,and effective plastic strain of selected points.

Fig.11.The relationship between microscopic morphology feature and dynamic mechanical response: (a) The contours of maximum stress-strain state of the steel plate; (b) The statistical method of samples; (c) The simulation results of each sample; (d) The defect sizes of each sample; (e) The scatter plot of samples; (f) The fitting models of samples.

Firstly,the values of the parameters of each sample were shown in Fig.11(c), the error bars showed the maximum and minimum values of the selected grids on each sample,some samples had large error bars(i.e.,the points in green circles),so it demonstrated that to select multiple grids and average them were necessary.The variations of above averaged parameters can be also recognized here.The average pressure decreased largely first from 0# to 3#sample,i.e.,6.26 to 0.81 GPa(0-4.8 cm from explosion center),and then became relatively smooth from 4# to 6# samples in the later stage, i.e., 0.47-0.38 GPa (4.8-8.4 cm from explosion center).Different from the pressure, the average effective stress changed little in the first three samples (0-3.6 cm from explosion center),but decreased obviously from 2#to 6#samples,i.e.0.70-0.58 GPa(3.6-8.4 cm from explosion center); Besides, the average effective plastic strain had the same variation trend with the average pressure,it reduced significantly in the first four samples,i.e.0.40-0.06,but decreased less in the later stages, i.e.0.03-0.02.Secondly, the quantitative research on defect area or diameter was also conducted.

From Fig.11(c) and 11(d), it can be recognized that the defect area had a similar trend with the average pressure, which represented that the mechanical factor was the main cause of the formation of microscopic defects.To explore the relationship between microscopic defect and mechanical factor of steel plate,the average pressure(x)and defect area(y)of each sample were plotted into a scatter plot(see Fig.11(e)).It can be observed that the pressure of 0# sample was the largest, and the corresponding defect area was also the largest.As the distance from the explosion center increased(i.e.,1# and 2#), the pressure of these samples reduced gradually,and the defect area also decreased continuously;Different from the above situation, 3# to 6# samples were relatively far from the explosion center, the pressure variation was small, and the damaged area was close.Besides, what is noteworthy is that the defect area between 2# and 3# samples was large (i.e., 49.98 μm2and 10.62 μm2), although the pressure difference between them was relatively small (i.e., 2.01 GPa and 0.81 GPa).This phenomenon corresponds to the transition region (shown in Fig.4(b)) on the surface of sample 2#observed by the stereoscopic microscope.For a better understanding of the relationship between microscopic defect and mechanical factor, the above two were fitted by the CPFM, the Monomolecular growth fitting model (MGFM), and the Logistic fitting model (LFM), respectively (see Fig.11(f)).From the fitting models, it can be recognized that there was no obvious difference in the fitting results of the last four samples by the three equations (see the purple area in Fig.11(f)).However, compared with MGFM and LFM,the CPFM can describe the law of defect area and average pressure better, especially at the 0# to 2# samples.Consequently,the CPFM(see Eq.(6))can be utilized as an effective way to predict the explosive damage of Q345 steel plate.

5.Conclusions

In this study, the explosive damage feature and dynamic mechanical response of the Q345 steel plate were investigated by using macroscopic and microscopic characterization techniques(i.e., SM and SEM) and dynamics simulation in combination.Combined with numerical simulation and microscopic defect characteristic data, a variety of mathematical equations corresponding to them were obtained by data fitting,and the best result(i.e.,CPFM)was finally selected to express the relationship between macroscopic dynamic mechanical response and microscopic defect of the steel plate under explosive load.The results of this study can be summarized as follow:

(1) The axial impact pressure of the explosive was large,and the directional gain was obvious.This caused the plastic deformation of Q345 steel plate was the largest around the area directly below the explosive, and the maximum displacement was 1.65 cm; The surface macroscopic damage characteristics were mainly the cave formation(with a maximum diameter of 2 mm), which was concentrated in the range of 0-3.0 cm from the explosion center.

(2) The surface microscopic defect of Q345 steel plate mainly includes two types:cavity and void.In the range of 0-2.5 cm,the variation of defect area can be fitted well by the CPFM,while at 2.5-7.5 cm, the EFM presented a good fitted effect.Besides, the overall trend of defect diameter can be effectively reflected by the EFM.Due to the steel plate melting and splashing under the high temperature and pressure, the shape of the cavity and void center were regular and smooth near the explosion center (0-2.5 cm), and irregular and rough far from the explosion center(2.5-7.5 cm).

(3) The numerical simulation showed that the pressure and effective stress of the first two samples (i.e., 0-2.4 cm from explosion center)increased significantly at the first 25 μs and then fell in the follow-up.While the rest samples (i.e.,2.4-8.4 cm from explosion center) rose relatively slowly.Besides, the effective stress of the surface of 0# and 1#sample changed obviously during the period of 80-120 μs,which may be caused by the pressure spread and reflected the bottom of the plate; The pressure, effective stress, and effective plastic strain were higher near the explosion center,but lower away from the explosion center.

(4) The defect formation of steel plate under the explosion could be related to its maximum stress-strain state.A corresponding relationship was established between the defect area of the surface and the pressure: within 6.26-2.01 GPa,the defect area gradually decreased from 85.30 to 49.98 μm2;However, When the pressure was in the range of 2.01-0.81 GPa,the defect area suddenly dropped from 49.98 to 10.62 μm2.Therefore, the critical pressure causing the plate defects may be approximately 2.01 GPa.The relation between defect area and pressure can be well fitted by CPFM.

This study provides an idea and method to study the relationship between microscopic defect and macroscopic dynamic mechanical response.On this basis,the influence of different explosive charges or different distances on the microstructure of different metal components can be explored in the future.

Declaration of competing interest

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.

Acknowledgements

The work was supported by Science and Technology Project of Fire Rescue Bureau of Ministry of Emergency Management (Grant No.2022XFZD05), S&T Program of Hebei (Grant No.22375419D),National Natural Science Foundation of China(Grant No.11802160).